A sensitivity analysis of the long-term expected utility of optimal portfolios
aa r X i v : . [ q -f i n . M F ] J un A sensitivity analysis of the long-term expected utilityof optimal portfolios
Hyungbin Park ∗ and Stephan Sturm † June 11, 2019
Abstract
This paper discusses the sensitivity of the long-term expected utility of optimal portfolios for an investor with constantrelative risk aversion. Under an incomplete market given by a factor model, we consider the utility maximization problemwith long-time horizon. The main purpose is to find the long-term sensitivity, that is, the extent how much the optimalexpected utility is affected in the long run for small changes of the underlying factor model. The factor model inducesa specific eigenpair of an operator, and this eigenpair does not only characterize the long-term behavior of the optimalexpected utility but also provides an explicit representation of the expected utility on a finite time horizon. We concludethat this eigenpair therefore determines the long-term sensitivity. As examples, explicit results for several market modelssuch as the Kim–Omberg model for stochastic excess returns and the Heston stochastic volatility model are presented. : 91G10, 93E20, 49L20, 60J60
JEL Classification : G11, C61
Keywords : Portfolio optimization, sensitivity analysis, spectral analysis, ergodic Hamilton–Jacobi–Bellman equation,Hansen–Scheinkman decomposition
Finding an optimal investment strategy is an important topic in mathematical finance. There are several ways to formulatethe optimal investment problem and one of the commonly accepted formulations is the use of utility function. An agentwants to maximize the expectation of the utility U by trading assets in a market. This paper also concerns this formulationof optimal expected utility, that is, sup Π ∈X E P (cid:2) U (Π T ) (cid:3) (1.1)for X the family of wealth processes of admissible portfolios.The analysis of this problem depends on the market completeness/incompleteness. The complete market case isrelatively easy to find the optimal expected utility (see Section 1.3), whereas the incomplete market case is more complicatedand requires advanced techniques. This paper deals with an incomplete market modeled by a factor model. Such factormodels are widely used in the quantitative finance literature. In the following we provide first an overview of the topicof the paper, review the relevant literature and present the relative straightforward case of a complete market given byone-dimensional diffusion model. The main purpose of this paper is to develop a sensitivity analysis of the long-term optimal expected utility. We considertwo kinds of sensitivities. The first is the sensitivity with respect to the initial factor, e.g., the current spot volatility if thefactor process is modeling the evolution of the volatility. For the initial value χ = X of the factor process, we study thebehavior of ∂∂χ sup Π ∈X E P (cid:2) U (Π T ) (cid:3) for large T. The second is the sensitivity with respect to a change in the drift or volatility function, e.g., reversion speed,mean reversion level and volatility of volatility for a mean-reverting volatility process. Let ǫ be a perturbation parameter ∗ Corresponding author. Department of Mathematical Sciences and RIMS, Seoul National University, 1, Gwanak-ro, Gwanak-gu, Seoul, Republicof Korea. Email: [email protected], [email protected] † Department of Mathematical Sciences, Worcester Polytechnic Institute, 100 Institute Rd, Worcester, MA, USA. Email: [email protected] nd consider a perturbed asset price S ǫ with S = S . Denote by X ǫ the family of wealth processes of admissible portfolioswith the perturbed asset model S ǫ . The precise meanings of S ǫ and X ǫ are discussed in Section 6 and 7.1. For the long-termsensitivity, we are interested in the behavior of ∂∂ǫ (cid:12)(cid:12)(cid:12) ǫ =0 sup Π ∈X ǫ E P (cid:2) U (Π T ) (cid:3) for large T. To achieve this, we combine several techniques: the duality approach (Kramkov and Schachermayer (1999)), thedynamic programming principle, the ergodic Hamilton–Jacobi–Bellman (HJB) equation (Knispel (2012)), the Hansen–Scheinkman decomposition (Hansen and Scheinkman (2009), Qin and Linetsky (2016)) and results on sensitivities forlong-term cash flows (Park (2018)). The asymptotic behavior of the sensitivities of (1.1) can be characterized by a solutionpair ( λ, φ ) of an ergodic HJB equation. Theorem 5.1 provides an exact representation of the optimal expected utility on afinite time horizon in terms of the asymptotic parameters ( λ, φ ) with a multiplicative error term. Besides being the maintool for the derivation of the results for the sensitivities, we believe this result is of interest on its own and might be of usefor further analysis. A precise formulation of the results and a detailed discussion on how the mentioned techniques canbe brought together to achieve these results will be given in Section 3.To make the objective of this paper clear and discern the problem at hand from similar problems, let us make theformulation we study precise: We consider the problemlim T →∞ T ∂∂ǫ (cid:12)(cid:12)(cid:12) ǫ =0 ln (cid:12)(cid:12)(cid:12) sup Π ∈X ǫ E P (cid:2) U (Π T ) (cid:3)(cid:12)(cid:12)(cid:12) for an investor with constant relative risk aversion larger than 1, i.e. utility function U ( x ) = x p p , p <
0. Specifically,we calculate the normalized asymptotic behavior of the derivative on a log scale. This is different from the problem tooptimize the long-term growth rate, where one optimizes over the normalized optimal growth rate on a logarithmic scaleand then analyzes its sensitivity. While both questions are economically meaningful, we focus in the current paper on thefirst type of sensitivity.
Many authors have worked on the optimal long-term investment problem: Fleming and McEneaney (1995) solve theoptimization problem of the long-term growth of expected utility for an investor with constant relative risk aversionby reformulating it as an infinite-time horizon risk-sensitive control problem. Guasoni and Robertson (2012) develops amethod to derive optimal portfolios explicitly in a general diffusion model of incomplete markets for an investor with powerutility. Liu and Muhle-Karbe (2013) explain how to compute optimal portfolios using stochastic control and convex duality.Special emphasis is placed on long-horizon asymptotics that lead to particularly tractable results. Robertson and Xing(2015) study the large time behavior of solutions to semi-linear Cauchy problems with quadratic gradients. Their analysishas direct applications to risk-sensitive control and long-term portfolio choice problems.Sensitivity analysis of optimal investment for fixed time horizon has also attracted many authors: Kramkov and Sˆırbu(2006) conduct a sensitivity analysis of the optimal expected utility with respect to a small change in initial capital or in aportfolio constraint. Larsen and ˇZitkovi´c (2007) investigate the stability of utility-maximization in complete and incompletemarkets under small perturbations. They identify the topologies on the parameter process space and the solution spaceunder which utility-maximization is a continuous operation. Backhoff and Silva (2017) conduct a first order sensitivityanalysis of some parameterized stochastic optimal control problems. Their main tool is the one-to-one correspondencebetween the adjoint states appearing in a weak form of the stochastic Pontryagin principle and the Lagrange multipliersassociated to the state equation. Larsen et al. (2018) study the first-order approximation for the power investor’s valuefunction and its second-order error is quantified in the framework of an incomplete financial market. Mostovyi and Sˆırbu(2017) investigate the sensitivity of the optimal expected utility in a continuous semimartingale market with respect tosmall changes in the market price of risk. For a general utility function, they derive a second-order expansion of thevalue function, a first-order approximation of the terminal wealth, and construct trading strategies. Mostovyi (2018)develops a sensitivity analysis for the expected utility maximization problem with respect to small perturbations in thenumeraire in an incomplete market model, where under an appropriate numeraire the stock price process is driven by asigma-bounded semimartingale. The author also establishes a second-order expansion of the value function and a first-order approximation of the terminal wealth. Monin and Zariphopoulou (2014) explore “portfolio Greeks,” which measurethe sensitivities of an investor’s optimal wealth to changes in cumulative excess stock return, time, and other marketparameters. Backhoff Veraguas and Silva (2018) study the issue of sensitivity with respect to model parameters for theproblem of utility maximization from final wealth in an incomplete Samuelson model for utility functions of positive-powertype by reformulating the maximization problem in terms of a convex-analytical support function of a weakly-compact set.This paper is closely related to and builds upon Park (2018) who investigates the long-term sensitivity of the expectation E h e − α R t θ ( S u ) du i or a perturbation of the underlying stochastic process S. In complete markets, we can use the result of his paper becausethe optimal expected utility can be expressed by an expectation of this form as we will see in Section 1.3. In incompletemarkets, however, the optimal expected utility cannot be expressed in the above form, thus one cannot rely on his result.We have to use in the current paper more advanced and complicated techniques to tackle the case of incomplete marketmodels.The current paper is structured as follows. In the remainder of Section 1 we discuss the case of a complete market modelas a warm up. Section 2 provides the model set up and specifies the market model and the optimization problem. Themain idea of this paper is presented in Section 3 using a heuristic argument. In Section 4, we display two examples: theKim–Omberg model and the Heston model. The dual formulation of the utility maximization problem and the Hansen–Scheinkman decomposition are discussed in Section 5 to provide rigorous results in the following: Sensitivity with respectto the initial factor is studied in Section 6, and those with respect to the drift and volatility are presented in Section 7.Section 8 summarizes the results of this paper. Proofs and detailed calculations are given in appendices.
As a warm up, this section discusses the long-term sensitivity of the optimal expected utility in a complete market as thisfollows easily from Park (2018). He investigates the long-term sensitivity of the expectation E h e − α R t θ ( S u ) du i for a real number α, a continuous function θ and an underlying asset process S . We show that the optimal expected utilityin a complete market can be expressed as this form of expectation, and so the results of Park (2018) directly applied.We consider the following market model: The price S of a risky asset (e.g., stock) satisfies dS t = b ( S t ) dt + ς ( S t ) dW t , S = s, with b and ς continuous functions, ς positive, such that this SDE has a unique non-explosive strong solution. Here, theprocess W is a Brownian motion under the physical probability measure P . Without loss of generality we assume that theshort interest rate is zero, so the market price of risk is θ t := θ ( S t ) = b ( S t ) ς ( S t ) . An investor with constant relative risk aversion 1 − p , p <
0, aims to maximize the expected power utility at the terminaltime U ( χ, T ) := sup Π ∈X E P (cid:2) U (Π T ) (cid:3) = 1 p inf Π ∈X E P (cid:2) Π pT (cid:3) . (1.2)By the homotheticity of power utility we can assume without loss of generality unit initial capital. Let P ∗ be the uniquerisk-neutral measure, and denote by L T the Radon–Nikod´ym derivative on F T , that is, L T = d P ∗ d P (cid:12)(cid:12)(cid:12) F T . It is known that the optimal investment portfolio value isˆΠ T = c T ( U ′ ) − ( L T )where c T is a constant determined by the budget constraint1 = E P ∗ (cid:2) ˆΠ T (cid:3) = c T E P ∗ (cid:2) ( U ′ ) − ( L T ) (cid:3) = c T E P ∗ h L / ( p − T i . Thus the optimal expected utility is E P (cid:2) U ( ˆΠ T ) (cid:3) = E P (cid:2) U (cid:0) c T U ′− ( L T ) (cid:1)(cid:3) = 1 p c pT E P (cid:2) L p/ ( p − T (cid:3) = 1 p E P (cid:2) L p/ ( p − T (cid:3) E P ∗ (cid:2) L / ( p − T (cid:3) p = 1 p E P ∗ (cid:2) L / ( p − T (cid:3) E P ∗ (cid:2) L / ( p − T (cid:3) p = 1 p E P ∗ (cid:2) L / ( p − T (cid:3) − p . This expectation can be expressed in terms of the market price of risk since the Radon–Nikod´ym derivative L T is L t = e − R t θ s dW s − R t θ s ds = e − R t θ s dW ∗ s + R t θ s ds , ≤ t ≤ T, ith W ∗ t := W t + R t θ s ds a P ∗ -Brownian motion. If we define a measure ˆ P from P ∗ with the Girsanov kernel − p θ t , then E P (cid:2) U (cid:0) ˆΠ T (cid:1)(cid:3) = 1 p E P ∗ (cid:2) L / ( p − T (cid:3) − p = 1 p E P ∗ h e − p R T θ s dW ∗ s + p − R T θ s ds i − p = 1 p · E ˆ P h e p − p )2 R T θ s ds i − p = 1 p E ˆ P h e − α R T θ s ds i − p with α := − p − p ) . The ˆ P -dynamics of S is dS t = (cid:16) b ( S t ) + pς ( S t ) θ ( S t )1 − p (cid:17) dt + ς ( S t ) d ˆ W t (1.3)where ˆ W is a ˆ P -Brownian motion. Thus the sensitivity analysis of the optimal expected utility boils down to the sensitivityanalysis of v ( s, T ) := E ˆ P h e − α R T θ ( S u ) du i (1.4)for s = S which can be done using the results of Hansen and Scheinkman (2009) and Park (2018) for the underlyingprocess (1.3).From the Hansen–Scheinkman decomposition, one can find an eigenvalue and an eigenfunction ( λ, φ ) (called the recur-rent eigenpair) of the pricing operator P T , defined by P T φ ( s ) = e − λT φ ( s )where P T φ ( s ) = E ˆ P h e − α R T θ ( S u ) du φ ( S T ) (cid:12)(cid:12)(cid:12) S = s i . They characterize the long-term behavior of v ( s, T ). Specifically, under some assumptions, the limitlim T →∞ v ( s, T ) e − λT φ ( s )exists and is independent of s. For the long-term initial-value sensitivity, one can show thatlim T →∞ ∂∂s ln v ( s, T ) = φ ′ ( s ) φ ( s ) . From this one can derive the actual sensitivity of the expected utility noting ∂∂s ln (cid:12)(cid:12) U ( s, T ) (cid:12)(cid:12) = ∂∂s ln (cid:16) − p v − p ( s, T ) (cid:17) = (1 − p ) ∂∂s ln v ( s, T ) . For the parameter sensitivity with respect to the drift and volatility, let ǫ be the perturbation parameter in the driftor volatility, and denote by v ǫ ( s, T ) the corresponding expectation in Eq.(1.4). Using the family of recurrent eigenpairs( λ ǫ , φ ǫ ) ǫ> , one can prove that lim T →∞ T ∂∂ǫ (cid:12)(cid:12)(cid:12) ǫ =0 ln v ǫ ( s, T ) = − ∂λ ǫ ∂ǫ (cid:12)(cid:12)(cid:12) ǫ =0 from which the sensitivity of the actual expected utility can be inferred as above by multiplying with 1 − p .We emphasize that the main line argument in this section cannot be applied to incomplete markets. Our reasoningrelies on the fact that in the complete market case the dual optimization problem is posed over a single risk-neutral measureand thus has a trivial solution. This cannot be generalized to a factor diffusion model describing an incomplete marketwhere we have an optimization problem over infinitely many risk-neutral measures.For the rest of this section, we investigate an example of a complete market as discussed above. We study the long-termsensitivity of the optimal expected utility when the underlying asset follows an Ornstein–Uhlenbeck process. This is oftenassumed when modeling commodities as gold, silver and oil. Assume that the asset follows dS t = ( µ − bS t ) dt + ς dW t , S = s, (1.5)under the physical measure and the short interest rate is zero. Then, the market price of risk is θ ( S t ) := µ − bS t ς , which connects the two Brownian motions, under the physical measure and under the risk-neutral measure, via dW ∗ t = dW t + θ ( S t ) dt. We want to analyze the value function v ( s, T ) given in Eq.(1.4). In this case the generator corresponding to the asset pricedynamics under ˆ P is given by L φ ( s ) = 12 ς φ ′′ ( s ) + µ − bs − p φ ′ ( s ) − αθ ( s ) φ ( s ) nd one can show that the recurrent eigenvalue λ and the recurrent eigenfunction φ of −L are λ = b ( √ − p − − p ) , φ ( s ) = e − As − Bs , where A = b ( √ − p − − p ) ς , B = − µ ( √ − p − − p ) ς . Theorem 1.1.
Under the Ornstein–Uhlenbeck model in Eq. (1.5) , the long-term sensitivities of the optimal expected utilityin Eq. (1.2) are lim T →∞ ∂∂s ln (cid:12)(cid:12) U ( s, T ) (cid:12)(cid:12) = (1 − p ) φ ′ ( s ) φ ( s ) = − (1 − p )( As + B ) , lim T →∞ T ∂∂µ ln (cid:12)(cid:12) U ( s, T ) (cid:12)(cid:12) = − (1 − p ) ∂λ∂µ = 0 , lim T →∞ T ∂∂b ln (cid:12)(cid:12) U ( s, T ) (cid:12)(cid:12) = − (1 − p ) ∂λ∂b = − √ − p − , lim T →∞ T ∂∂ς ln (cid:12)(cid:12) U ( s, T ) (cid:12)(cid:12) = − (1 − p ) ∂λ∂ς = 0 . The model setup of the current paper is as follows: Let (Ω , F , ( F t ) t ≥ , P ) be the canonical path space of a two-dimensionalBrownian motion ( W ,t , W ,t ) t ≥ . The filtration ( F t ) t ≥ is the usual completion of the natural filtration of ( W ,t , W ,t ) t ≥ . The measure P is referred to as the physical measure. The dynamics of the risky asset is given by the following stochasticdifferential equations (SDEs) dS t = b ( X t ) S t dt + ς ( X t ) S t dW ,t , S = 1 , (2.1) dX t = m ( X t ) dt + σ ( X t ) dW ,t + σ ( X t ) dW ,t , X = χ, (2.2)which is a typical way to define a stochastic factor model. The processes S and X describe an asset price and its underlyingfactor process, respectively. The five functions m, σ , σ , b, ς and the real number χ satisfy the following assumptions.Let ( ℓ, r ) be an open interval in R for −∞ ≤ ℓ < r ≤ ∞ . A 1.
Let χ ∈ ( ℓ, r ) and let m, σ , σ be continuous functions on ( ℓ, r ) such that σ + σ > . The SDE (2.2) has a uniquenon-explosive (i.e., P [ X t ∈ ( ℓ, r ) for all t ≥
0] = 1 ) strong solution X . A 2.
The functions b, ς are continuous and ς is strictly positive on ( ℓ, r ) . Under these assumptions the asset price process is well-defined and can be written as S t = e R t ( b − ς )( X s ) ds + R t ς ( X s ) dW ,s . A 3.
For each fixed time T, there exists a probability measure on F T such that the discounted asset price process is a localmartingale on [0 , T ] . It is well-known that this assumption is equivalent to the absence of arbitrage in the market in the sense of no free lunchwith vanishing risk (Delbaen and Schachermayer (1994)).Without loss of generality we will assume that the short interest rate is zero so that the value of the money marketaccount is one at all time t. The market price of risk is then given by θ t = θ ( X t ) = b ( X t ) ς ( X t ) . (2.3)An investor wants to maximize the expected utility of the value of their portfolio at terminal time T by trading the assetand the money market account. A portfolio is a predictable processes ψ which is S -integrable. The value process Π = Π ψ of the portfolio ψ is Π t = Π + Z t ψ u dS u , ≤ t ≤ T. We denote by X the family of nonnegative value processes with initial wealth Π equal to 1, that is, X = { Π ψ ≥ ψ is a portfolio and Π ψ = 1 } . (2.4) he investor is assumed to have constant relative risk aversion 1 − p >
1, i.e., the utility function corresponding to theirpreferences is of negative power type U ( x ) = x p p , p < . For given initial capital, the goal of the investor is to maximize the expected value at the terminal wealth, that is,sup Π ∈X E P (cid:2) U (Π T ) (cid:3) = 1 p inf Π ∈X E P (cid:2) Π pT (cid:3) . (2.5)Without loss of generality we can assume that the initial capital is equal to one, thanks to the homotheticity of theinvestor’s preferences. The main purpose of the current paper is to investigate two types of long-term sensitivity with respect to the perturbationof S and X. One is the sensitivity with respect to the initial value χ = X of the factor process (2.2), ∂∂χ ln (cid:12)(cid:12)(cid:12) sup Π ∈X E P [ U (Π T )] (cid:12)(cid:12)(cid:12) . (3.1)The other type concerns the sensitivities with respect to the five functions m, σ , σ , b, ς. Let m ǫ , σ ,ǫ , σ ,ǫ , b ǫ , ς ǫ beperturbed functions with perturbation parameter ǫ (for a precise definition, see Section 7.1). Denote by S ǫ the perturbedasset process induced by these perturbed functions, and consider the family X ǫ of wealth processes given by Eq.(2.4)generated by the perturbed asset process S ǫ . The sensitivity of interest is that with respect to the ǫ -perturbation, ∂∂ǫ (cid:12)(cid:12)(cid:12) ǫ =0 ln (cid:12)(cid:12)(cid:12) sup Π ∈X ǫ E P [ U (Π T )] (cid:12)(cid:12)(cid:12) . Remark 3.1.
We note that the assumption S = 1 in Eq. (2.1) does not restrict the generality of the results. In fact, inthe factor model, the optimal expected utility is independent of the initial value of the stock price as the stock dynamicsscale linearly. This is in contrast to the results for the complete market case in Section 1.3, as there also drift and volatilityfunctions depend on the stock price. In the following, we will present the main ideas how to derive the long-term initial-factor sensitivity by surveying theessential steps of the argument. The technical details are relegated to Section 6.(i) From the dual formulation of utility maximization problem (Kramkov and Schachermayer (1999), details will besurveyed in Section 5.1), we know that U ( χ, T ) := sup Π ∈X E [ U (Π T )] = 1 p (cid:16) E P (cid:2) ˆ Y qT (cid:3)(cid:17) − p (3.2)for some nonnegative supermartingale ˆ Y and q := − p/ (1 − p ); define v ( χ, T ) := E P (cid:2) ˆ Y qT (cid:3) = E P (cid:2) ˆ Y qT (cid:12)(cid:12) X = χ (cid:3) . (ii) The sensitivity in Eq.(3.1) is ∂∂χ ln (cid:12)(cid:12)(cid:12) sup Π ∈X E P (cid:2) U (Π T ) (cid:3)(cid:12)(cid:12)(cid:12) = (1 − p ) ∂∂χ ln v ( χ, T ) , so it suffices to evaluate the long-term behavior of ∂∂χ ln v ( χ, T ) . (iii) The function v satisfies a HJB equation (details are given in Section 5.1).(iv) The function v can be approximated by a solution pair ( λ, φ ) of an ergodic HJB equation (see Eq.(5.8)) in the sensethat e − λT φ ( χ ) is asymptotically equal to v ( χ, T ) up to a constant factor, that is, v ( χ, T ) ≃ e − λT φ ( χ )(where we use the notation f T ≃ g T to denote that the limit lim T →∞ f T g T for two positive functions f T and g T converges to a positive constant). To derive this result, we rely on the HJB representation of v derived in (iii). v) By taking the partial derivative to the above asymptotics, one can anticipate that ∂∂χ ln v ( χ, T ) ≃ φ ′ ( χ ) φ ( χ ) (3.3)and this is indeed one of the main results of this paper and is stated in detail in Theorem 3.2. This approach ismotivated by Section 3 in Park (2018).(vi) To make this asymptotic result rigorous, one needs to control the error terms. This can be done using a probabilisticrepresentation of the function v , v ( χ, T ) = e − λT φ ( χ ) E Q h φ ( X T ) e R T f ( X s ,s ; T ) ds i , for a probability measure Q and a continuous function f. The precise result is given in Theorem 5.1, the proof relies onan adaption of the Hansen–Scheinkman decomposition to the current context. Thus, by taking the partial derivativedirectly, we get ∂∂χ ln v ( χ, T ) = φ ′ ( χ ) φ ( χ ) + ∂∂χ ln E Q h φ ( X T ) e R T f ( X s ,s ; T ) ds i . Under reasonable conditions the error term ∂∂χ ln E Q h φ ( X T ) e R T f ( X s ,s ; T ) ds i goes to zero as T → ∞ and we obtain Eq.(3.3), the desired result.The following theorem is the main result on the sensitivity with respect to the initial-factor. The proof will be given inSection 6. Theorem 3.2.
Assume A1 – 10 (stated in Sections 2 and 5.1) and additionally that the map χ E Q h φ ( X T ) e R T f ( X s ,s ; T ) ds (cid:12)(cid:12)(cid:12) X = χ i is continuously differentiable with derivative converging to zero as T → ∞ . Then lim T →∞ ∂∂χ ln v ( χ, T ) = φ ′ ( χ ) φ ( χ ) . (3.4) Remark 3.3.
This result is very similar in spirit to the results by (Robertson and Xing, 2015, Eq.(1.4) and Theorem2.11). They also discuss asymptotic behavior of the type as Eq. (3.4) . Their approach as well as the assumptions neededare however different from the current paper.
For the second topic of the paper, the sensitivities with respect to small perturbation parameters, we proceed in thesame way and provide an overview of the main steps of the argument; the technical details will be given at Section 7.(i’) – (iv’) For each ǫ, we can follow the approach of the sensitivity analysis with respect to the initial factor. Specificallyconducting steps (i) – (iv) as above and defining v ǫ ( χ, T ) and ( λ ǫ , φ ǫ ) accordingly, we obtain v ǫ ( χ, T ) ≃ e − λ ǫ T φ ǫ ( χ ) . (v’) By taking the partial derivative to the above asymptotics, we have1 T ∂∂ǫ (cid:12)(cid:12)(cid:12) ǫ =0 ln v ǫ ( χ, T ) ≃ − ∂∂ǫ (cid:12)(cid:12)(cid:12) ǫ =0 λ ǫ . (3.5)(vi’) The function v ǫ ( x, T ) has the probabilistic representation v ǫ ( χ, T ) = e − λ ǫ T φ ǫ ( χ ) E Q ǫ h φ ǫ ( X ǫT ) e R T f ǫ ( X ǫs ,s ; T ) ds i . Thus, by taking the partial derivative, it follows that1
T ∂∂ǫ (cid:12)(cid:12)(cid:12) ǫ =0 ln v ǫ ( χ, T ) = − ∂λ ǫ ∂ǫ (cid:12)(cid:12)(cid:12) ǫ =0 + 1 T ∂∂ǫ (cid:12)(cid:12)(cid:12) ǫ =0 ln φ ǫ ( χ ) + 1 T ∂∂ǫ (cid:12)(cid:12)(cid:12) ǫ =0 ln E Q ǫ h φ ǫ ( X ǫT ) e R T f ǫ ( X ǫs ,s ; T ) ds i . The second term goes to zero as T → ∞ and under reasonable conditions also the error term1 T ∂∂ǫ (cid:12)(cid:12)(cid:12) ǫ =0 ln E Q ǫ h φ ǫ ( X ǫT ) e R T f ǫ ( X ǫs ,s ; T ) ds i vanishes as T → ∞ , thus we obtain Eq.(3.5). heorem 3.4. Assume B1 – 2, conditions (i) – (iii) in Theorem 7.1 and additionally that the map ǫ E Q ǫ h φ ( X ǫT ) e R T f ( X ǫs ,s ; T ) ds i is continuously differentiable at ǫ = 0 with T ∂∂ǫ (cid:12)(cid:12)(cid:12) ǫ =0 E Q ǫ h φ ( X ǫT ) e R T f ( X ǫs ,s ; T ) ds i converging to zero as T → ∞ . Then lim T →∞ T ∂∂ǫ (cid:12)(cid:12)(cid:12) ǫ =0 ln v ǫ ( χ, T ) = − ∂λ ǫ ∂ǫ (cid:12)(cid:12)(cid:12) ǫ =0 . Before implementing the sketched program rigorously, we want to show in this section which results can actually be achievedin specific examples. The power of our approach is demonstrated by deriving explicit formulas for the Kim–Omberg modelof stochastic excess returns and the Heston stochastic volatility model.
In the Kim–Omberg model (Kim and Omberg (1996)) the asset price S and the stochastic excess returns X satisfy dS t = µX t S t dt + ςS t dW ,t , S = 1 ,dX t = k ( m − X t ) dt + σ dZ t , X = χ (4.1)for correlated Brownian motions W and Z with correlation parameter ρ ∈ ( − , k, the volatilities ς, σ are positive and the return µ, the mean reversion level m are real numbers.This fits into the standard model by setting σ = ρσ, σ = p − ρ σ and W ,t = √ − ρ Z t − ρ √ − ρ W ,t so that dX t = k ( m − X t ) dt + σ dW ,t + σ dW ,t , X = χ. The market price of risk is given as θ t := µς X t . Define α = k + qµσ ς , α = σ + σ − q , α = km, α = q α + q (1 − q ) α µ /ς . and B = α − α α , C = α ( α − α ) α α for q being the dual exponent of the utility function, q = − p − p . Then the recurrent eigenpair is λ = − α C + α C + 12 σ B,φ ( x ) = e − Bx − Cx . Theorem 4.1.
In the Kim–Omberg model, assume the parameters satisfy k + qµρσς + Bσ > . (4.2) Then the long-term sensitivities of the optimal expected utility in Eq. (3.2) are given by lim T →∞ ∂∂χ ln (cid:12)(cid:12) U ( s, T ) (cid:12)(cid:12) = − (1 − p )( Bχ + C ) , lim T →∞ T ∂∂k ln (cid:12)(cid:12) U ( s, T ) (cid:12)(cid:12) = (1 − p ) α (cid:16) mα − α + α α (cid:17) C − (1 − p ) (cid:16) m − α ( α + α ) α (cid:17) C + (1 − p ) σ B α , lim T →∞ T ∂∂m ln (cid:12)(cid:12) U ( s, T ) (cid:12)(cid:12) = α α (1 − p ) kC − − p ) kC, lim T →∞ T ∂∂µ ln (cid:12)(cid:12) U ( s, T ) (cid:12)(cid:12) = − pσα α ( ρςα − kρςα − µσα ) ς α α − pσ ( ρςα − qρς − µσ )2 ς α α , im T →∞ T ∂∂ς ln (cid:12)(cid:12) U ( s, T ) (cid:12)(cid:12) = pqµ σ α α ( ρςα − kρςα − µσα ) ς α α + pµσ ( ρςα − kρς − µσ )2 ς α α lim T →∞ T ∂∂ρ ln (cid:12)(cid:12) U ( s, T ) (cid:12)(cid:12) = − α pC (cid:16) ( k − α ) µσςα ( α − α ) + ρσ (1 − q ) α − kµσςα (cid:17) + α pC (cid:16) ( k − α ) µσςα ( α − α ) + 2 ρσ (1 − q ) α − kµσςα (cid:17) + 12 pσ B (cid:16) ( k − α ) µσςα ( α − α ) + 2 ρσ (1 − q ) α (cid:17) , lim T →∞ T ∂∂σ ln (cid:12)(cid:12) U ( s, T ) (cid:12)(cid:12) = α (cid:16) pµ ( ρςα − kρς − µσ ) ς α ( α − α ) − − pσ + pµ ( kρς + µσ ) ς α (cid:17) C − α (cid:16) pµ ( ρςα − kρς − µσ ) ς α ( α − α ) − − p ) σ + pµ ( kρς + µσ ) ς α (cid:17) C − pµσ (cid:16) ρςα − kρς − µσς α ( α − α ) (cid:17) B. The proof of these asymptotic results can be found at Appendix D.
In the Heston stochastic volatility model (Heston (1993)) the asset price S and the stochastic variance process X satisfy dS t = µX t S t dt + ς √ X t S t dW ,t , S = 1 ,dX t = k ( m − X t ) dt + σ √ X t dZ t , X = χ for correlated Brownian motions W and Z with correlation parameter ρ ∈ ( − , k, the mean reversion level m, the volatilities ς, σ are positive, and the return µ is a real number. Assume the Fellercondition 2 km > σ , which ensures that the zero boundary of X is inaccessible.This fits into the standard model by setting σ = ρσ, σ = p − ρ σ and W ,t = √ − ρ Z t − ρ √ − ρ W ,t so that dX t = k ( m − X t ) dt + σ √ X t dW ,t + σ √ X t dW ,t , X = χ. The market price of risk is θ t := µς √ X t . Define β := k + qµρσς , β := q β + q (1 − qρ ) µ σ /ς , and B = (1 − q )( β − β )(1 − qρ ) σ . Then the recurrent eigenpair is λ = kmB, φ ( x ) = e − Bx . Theorem 4.2.
In the Heston model, assume the Feller condition km > σ and k + qµρσς > . (4.3) Then the long-term sensitivities of the optimal expected utility in Eq. (3.2) are lim T →∞ ∂∂χ ln (cid:12)(cid:12) U ( s, T ) (cid:12)(cid:12) = − (1 − p ) B, lim T →∞ T ∂∂k ln (cid:12)(cid:12) U ( s, T ) (cid:12)(cid:12) = (1 − p ) mB ( kβ − , lim T →∞ T ∂∂m ln (cid:12)(cid:12) U ( s, T ) (cid:12)(cid:12) = − (1 − p ) kB, lim T →∞ T ∂∂µ ln (cid:12)(cid:12) U ( s, T ) (cid:12)(cid:12) = kmq ( ρςβ − kρς − µσ )(1 − qρ ) σς β , lim T →∞ T ∂∂ς ln (cid:12)(cid:12) U ( s, T ) (cid:12)(cid:12) = kmpµσB ( ρςβ − ρkς − µσ ) ς β ( β − β ) , lim T →∞ T ∂∂ρ ln (cid:12)(cid:12) U ( s, T ) (cid:12)(cid:12) = kmB (cid:16) − pµσ ( β − k ) ςβ ( β − β ) + 2 pρ − qρ (cid:17) , lim T →∞ T ∂∂σ ln (cid:12)(cid:12) U ( s, T ) (cid:12)(cid:12) = kmB (cid:16) − p ) σ + pµ ( kρς + µσ − ρςβ ) ς β ( β − β ) (cid:17) . he proof of these asymptotic results can be found at Appendix E. Remark 4.3.
The conditions in Eq. (4.2) and Eq. (4.3) are there to guarantee that the process X is still mean-revertingunder the measures relevant for the analysis (details are discussed in the Appendices D and E). This condition is in spiritsimilar to the conditions one finds in the long term analysis of implied volatility in these models where the asymptoticregime depends on the mean-reversion property under the share measure (see, e.g., (Forde and Jacquier, 2011, Theorem2.1) and (Keller-Ressel, 2011, Section 6.1). We provide a mathematical background for the heuristic argument given in Section 3. First we discuss the dual formulationof the utility maximization problem and its characterization via the solution of an HJB equation. Then we introducethe ergodic HJB equation who can characterize the long-run problem and analyze it in terms of its eigenpair. Finally wegeneralize the Hansen–Scheinkmann decomposition to functionals of time-inhomogeneous Markov process to lay the groundfor the following sensitivity analysis. On the way we make precise the assumptions that are needed for our conclusions.
One of the main ideas is to employ the dual formulation of the utility maximization problem as presented in Kramkov and Schachermayer(1999). We recall (see Eq.(2.5)) the primal problem of utility maximization issup Π ∈X E P (cid:2) U (Π T ) (cid:3) This primal problem is related to the following dual formulation, which is a minimization probleminf Y ∈Y E P (cid:2) V ( Y T ) (cid:3) = inf Y ∈Y E P (cid:2) − Y qT /q (cid:3) , (5.1)where q = − p − p is the conjugate exponent of p and V ( y ) = − y q q is the dual conjugate of the utility function U. Here, Y is the family of nonnegative semimartingales Y with Y = 1 such that the product ( X t Y t ) t ≥ is a supermartingale forany X ∈ X . Denote by ˆ Y the optimal element in Y (Theorem 2.2 in Kramkov and Schachermayer (1999) guarantees theexistence of this optimum) and define v ( χ, T ) := E P (cid:2) ˆ Y qT (cid:3) = E P (cid:2) ˆ Y qT (cid:12)(cid:12) X = χ (cid:3) . (5.2)We emphasize that here χ is the initial value of the factor process. Note that the function v is not the actual dual valuefunction but a constant multiple of it. This follows from normalizing the dual initial condition which can be done thanksto the homotheticity of the power function y q . From Eq.(4.10) in Larsen et al. (2018), we knowsup Π ∈X E (cid:2) U (Π T ) (cid:3) = 1 p v − p ( χ, T ) (5.3)so that the long-term growth rate of the optimal expected utility in Eq.(5.11) islim T →∞ T ln (cid:12)(cid:12)(cid:12) sup Π ∈X E P (cid:2) U (Π T ) (cid:3)(cid:12)(cid:12)(cid:12) = (1 − p ) lim T →∞ T ln v ( χ, T ) . Under some conditions, we can characterize the function v as a solution of a HJB equation v t = 12 (cid:0) σ ( x ) + σ ( x ) (cid:1) v xx + sup ξ ∈ R (cid:8) l ( ξ, x ) v + h ( ξ, x ) v x (cid:9) , v ( x,
0) = 1 (5.4)where l ( ξ, x ) := − q − q ) (cid:0) θ ( x ) + ξ (cid:1) h ( ξ, x ) := m ( x ) − qθ ( x ) σ ( x ) − qξσ ( x ) . (5.5)Moreover, the optimal element ˆ Y ∈ Y of Eq.(5.1) can be expressed asˆ Y t = e − R t θ ( X s ) dW ,s − R t θ ( X s ) ds − R t ˆ ξ ( X s ,s ; T ) dW ,s − R t ˆ ξ ( X s ,s ; T ) ds , ≤ t ≤ T, (5.6)where θ ( X t ) := b ( X t ) ς ( X t ) is the market price of risk andˆ ξ ( x, t ; T ) := − σ ( x ) v x ( x, T − t )(1 − q ) v ( x, T − t ) (5.7) s the optimal control of the HJB equation (5.4). Under appropriate conditions the function v can be approximated usinga solution pair ( λ, φ ) of − λφ ( x ) = 12 (cid:0) σ ( x ) + σ ( x ) (cid:1) φ xx + sup ξ ∈ R (cid:8) l ( ξ, x ) φ + h ( ξ, x ) φ x (cid:9) (5.8)which is called the ergodic HJB equation. It is noteworthy that the real number λ and the function φ can be regarded asan eigenvalue and an eigenfunction of the operator −L where L φ = 12 (cid:0) σ ( x ) + σ ( x ) (cid:1) φ xx + sup ξ ∈ R (cid:8) l ( ξ, x ) φ + h ( ξ, x ) φ x (cid:9) . We will review the motivation of these arguments and the derivation in Appendix A.We make the following assumptions on the function v and the structure of the optimal element ˆ Y ∈ Y of the dualproblem without going into further details. For sufficient conditions and a more detailed discussion we refer to (Knispel,2012, p. 10–12), (Hern´andez-Hern´andez and Schied, 2006, Section 4) and (Kaise and Sheu, 2009, Sections 3 and 5). A 4.
The function v ( x, t ) given by (5.2) is twice continuously differentiable in x and once in t and satisfies the PDE (5.4) . A 5.
The optimizer ˆ Y of Eq. (5.1) is given by Eq. (5.6) . A 6.
There exist a real number λ and a continuously twice-differentiable positive function φ satisfying Eq. (5.8) such that v ( x, t ) e − λt φ ( x ) → C as t → ∞ for a positive constant C not depending on x. We can represent the function v in a simpler way. From Eq.(5.2), Eq.(5.6) and A5, it follows that v ( χ, T ) := E P (cid:2) ˆ Y qT (cid:3) = E P (cid:2) e − q R T θ ( X s ) dW ,s − q R T θ ( X s ) ds − q R T ˆ ξ ( X s ,s ; T ) dW ,s − q R T ˆ ξ ( X s ,s ; T ) ds (cid:3) = E ˆ P (cid:2) e − q (1 − q ) R T ( θ ( X s )+ˆ ξ ( X s ,s ; T )) ds (cid:3) (5.9)where ˆ P is a measure on F T defined as d ˆ P d P = E (cid:16) − q Z · θ ( X s ) dW ,s − q Z · ˆ ξ ( X s , s ; T ) dW ,s (cid:17) T (5.10)under A7 stated below. The ˆ P -dynamics of X is dX t = ( m ( X t ) − qθ ( X t ) σ ( X t ) − q ˆ ξ ( X t , t ; T ) σ ( X t )) dt + σ ( X ) d ˆ W ,t + σ ( X t ) d ˆ W ,t for a ˆ P -Brownian motion ( ˆ W ,t , ˆ W ,t ) . A 7.
For the function ˆ ξ given by Eq. (5.7) , the local martingale (cid:18) E (cid:16) − q Z · θ ( X s ) dW ,s − q Z · ˆ ξ ( X s , s ; T ) dW ,s (cid:1) t (cid:19) ≤ t ≤ T is a true martingale under the measure P . The solution pair ( λ, φ ) describes the long-term behavior of v ( χ, T ) for initial factor χ and maturity T as T → ∞ . Thelong-term growth rate of the optimal expected utility is defined aslim T →∞ T ln (cid:12)(cid:12)(cid:12) sup Π ∈X E P (cid:2) U (Π T ) (cid:3)(cid:12)(cid:12)(cid:12) (5.11)and can be described by the eigenvalue λ since − λ = lim T →∞ T ln v ( χ, T ) = 11 − p lim T →∞ T ln (cid:12)(cid:12)(cid:12) sup Π ∈X E P (cid:2) U (Π T ) (cid:3)(cid:12)(cid:12)(cid:12) , which follows from Eq.(5.3). The optimal control of the ergodic HJB equation (5.8) is a function of x, so we denote by ξ ∗ ( x ) . It is easy to check that ξ ∗ is given by ξ ∗ ( x ) = − σ ( x ) φ x ( x )(1 − q ) φ ( x ) (5.12)and Eq.(5.8) becomes − λφ ( x ) = 12 (cid:0) σ ( x ) + σ ( x ) (cid:1) φ xx + h ( ξ ∗ ( x ) , x ) φ x + l ( ξ ∗ ( x ) , x ) φ. (5.13)The long-term growth rate can be calculated as − λ = lim T →∞ T ln E ˆ P h e R T l ( ξ ∗ ( X s ) ,X s ) ds i . .2 Hansen–Scheinkman decomposition This section is inspired by the Hansen–Scheinkman decomposition in Hansen and Scheinkman (2009) and is adapted tothe current context. They study the decomposition of a multiplicative functional of a time-homogeneous Markov processinto a product of an exponential of the eigenvalue, the eigenfunction and an error term. In this section, we adapt theirmethod to a time-inhomogeneous Markov case. Recall from Eq.(5.9) that v ( χ, T ) = E ˆ P h e − q (1 − q ) R T (cid:0) θ ( X s )+ˆ ξ ( X s ,s ; T ) (cid:1) ds (cid:12)(cid:12)(cid:12) X = χ i (5.14)and the ˆ P -dynamics of X is dX t = ( m ( X t ) − qθ ( X t ) σ ( X t ) − q ˆ ξ ( X t , t ; T ) σ ( X t )) dt + σ ( X ) d ˆ W ,t + σ ( X t ) d ˆ W ,t for 0 ≤ t ≤ T and a two-dimensional ˆ P -Brownian motion ( ˆ W ,t , ˆ W ,t ) . We assume the following condition:
A 8.
For the functions ˆ ξ given by Eq. (5.7) and ξ ∗ given by Eq. (5.12) , the local martingale (cid:18) E (cid:16) q Z · ˆ ξ ( X s , s ; T ) − ξ ∗ ( X s ) d ˆ W ,s (cid:17) t (cid:19) ≤ t ≤ T is a true martingale under the measure ˆ P . Define a new measure ˜ P T on F T by d ˜ P T d ˆ P (cid:12)(cid:12)(cid:12) F T = E (cid:16) q Z · ˆ ξ ( X s , s ; T ) − ξ ∗ ( X s ) d ˆ W ,s (cid:17) T . (5.15)For simplicity we drop the subscript T and write just ˜ P . The ˜ P -dynamics of X is dX t = (cid:0) m ( X t ) − qθ ( X t ) σ ( X t ) − qξ ∗ ( X t ) σ ( X t ) (cid:1) dt + σ ( X t ) d ˜ W ,t + σ ( X t ) d ˜ W ,t with two-dimensional ˆ P -Brownian motion (cid:18) d ˜ W ,t d ˜ W ,t (cid:19) = (cid:18) qξ ∗ ( X t ) − q ˆ ξ ( X t , t ; T ) (cid:19) dt + (cid:18) d ˆ W ,t d ˆ W ,t (cid:19) . From Eq.(5.14) it follows that v ( χ, T ) = E ˆ P h e − q (1 − q ) R T ( θ ( X s )+ ξ ∗ ( X s )) ds e − q (1 − q ) R T (ˆ ξ ( X s ,s ; T ) − ξ ∗ ( X s )) ds i = E ˜ P h e − q (1 − q ) R T ( θ ( X s )+ ξ ∗ ( X s )) ds e − q (1 − q ) R T (ˆ ξ ( X s ,s ; T ) − ξ ∗ ( X s )) ds d ˆ P d ˜ P i . Define M t := φ ( X t ) φ ( χ ) e λt − q (1 − q ) R t ( θ ( X s )+ ξ ∗ ( X s )) ds , ; 0 ≤ t ≤ T. Then applying the Itˆo formula to M and using the ergodic HJB equation (5.13), it can be checked that M t = E (cid:16)Z · φ ′ ( X s ) φ ( X s ) σ ( X s ) d ˜ W ,s + Z · φ ′ ( X s ) φ ( X s ) σ ( X s ) d ˜ W ,s (cid:17) t , ≤ t ≤ T, and thus a ˜ P -local martingale. A 9.
With the solution pair ( λ, φ ) of A6, the ˜ P -local martingale ( M t ) ≤ t ≤ T is a true martingale. We use this random variable M T as a Radon–Nikod´ym derivative to defined a new measure P , that is d P d ˜ P (cid:12)(cid:12)(cid:12) F T = M T . (5.16)This measure P depends on T, but we suppress in the notation the dependence on T as before. Then v ( χ, T ) = e − λT φ ( χ ) E ˜ P h M T φ ( X T ) e − q (1 − q ) R T (ˆ ξ ( X s ,s ; T ) − ξ ∗ ( X s )) ds d ˆ P d ˜ P i = e − λT φ ( χ ) E P h φ ( X T ) e − q (1 − q ) R T (ˆ ξ ( X s ,s ; T ) − ξ ∗ ( X s )) ds d ˆ P d ˜ P i . he process (cid:18) dW ,t dW ,t (cid:19) = − φ ′ ( X t ) φ ( X t ) σ ( X t ) φ ′ ( X t ) φ ( X t ) σ ( X t ) ! dt + (cid:18) d ˜ W ,t d ˜ W ,t (cid:19) is a Brownian motion under P and the P -dynamics of X is dX t = (cid:16) m ( X t ) − qθ ( X t ) σ ( X t ) − qξ ∗ ( X t ) σ ( X t ) + φ ′ ( X t ) φ ( X t ) (cid:0) σ ( X t ) + σ ( X t ) (cid:1)(cid:17) dt + σ ( X t ) dW ,t + σ ( X t ) dW ,t . We now perform another change of measure to express the function v ( χ, T ) in a more manageable way. Before doingso, we express the Radon–Nikod´ym derivative d ˆ P d ˜ P in a different way to facilitate the calculation: d ˆ P d ˜ P = e q R T ξ ∗ ( X s ) − ˆ ξ ( X s ,s ; T ) d ˆ W ,s + q R T ( ξ ∗ ( X s ) − ˆ ξ ( X s ,s ; T )) ds = e q R T ξ ∗ ( X s ) − ˆ ξ ( X s ,s ; T ) d ˜ W ,s − q R T ( ξ ∗ ( X s ) − ˆ ξ ( X s ,s ; T )) ds = e q R T ξ ∗ ( X s ) − ˆ ξ ( X s ,s ; T ) dW ,s − q R T ( ξ ∗ ( X s ) − ˆ ξ ( X s ,s ; T )) ds + q R T ( ξ ∗ ( X s ) − ˆ ξ ( X s ,s ; T )) φ ′ ( Xs ) φ ( Xs ) σ ( X s ) ds . We will need an additional assumption that the argument works:
A 10.
The local martingale (cid:18) E (cid:18) q Z · ξ ∗ ( X s ) − ˆ ξ ( X s , s ; T ) dW ,s (cid:19) t (cid:19) ≤ t ≤ T is a true martingale under the measure P . We now define a new measure Q by d Q d P = E (cid:18) q Z · ξ ∗ ( X s ) − ˆ ξ ( X s , s ; T ) dW ,s (cid:19) T . (5.17)This measure Q depends on T, but we suppress in the notation the dependence on T as before. Then v ( χ, T )= e − λT φ ( χ ) E P h φ ( X T ) e − q (1 − q ) R T (ˆ ξ ( X s ,s ; T ) − ξ ∗ ( X s )) ds e q R T ( ξ ∗ ( X s ) − ˆ ξ ( X s ,s ; T )) φ ′ ( Xs ) φ ( Xs ) σ ( X s ) ds d Q d P i = e − λT φ ( χ ) E Q h φ ( X T ) e − q (1 − q ) R T (ˆ ξ ( X s ,s ; T ) − ξ ∗ ( X s )) ds e q R T ( ξ ∗ ( X s ) − ˆ ξ ( X s ,s ; T )) φ ′ ( Xs ) φ ( Xs ) σ ( X s ) ds i = e − λT φ ( χ ) E Q h φ ( X T ) e − q (1 − q ) R T ( ξ ∗ ( X s ) − ˆ ξ ( X s ,s ; T )) ds i . (5.18)For the last equality, we used Eq.(5.12). The Q -dynamics of X is dX t = (cid:16) m ( X t ) − qθ ( X t ) σ ( X t ) − q ˆ ξ ( X t , t ; T ) σ ( X t ) + φ ′ ( X t ) φ ( X t ) (cid:0) σ ( X t ) + σ ( X t ) (cid:1)(cid:17) dt + σ ( X t ) dB ,t + σ ( X t ) dB ,t (5.19)where (cid:18) dB ,t dB ,t (cid:19) = (cid:18) ξ ( X t , t ; T ) − ξ ∗ ( X t ) (cid:19) dt + (cid:18) dW ,t dW ,t (cid:19) is a Q -Brownian motion.In conclusion, we can express the function v ( χ, T ) and the dynamics of X in a simpler way. The following theoremfollows from Eq.(5.18) and Eq.(5.19). Theorem 5.1.
Assume A1 – 10. Then the function v ( χ, T ) can be decomposed as v ( χ, T ) = e − λT φ ( χ ) E Q h φ ( X T ) e R T f ( X s ,s ; T ) ds i (5.20) and the Q -dynamics of X is dX t = κ ( X t , t ; T ) dt + σ ( X t ) dB ,t + σ ( X t ) dB ,t , ≤ t ≤ T, where f ( x, t ; T ) = − q − q ) (cid:0) ξ ∗ ( x ) − ˆ ξ ( x, t ; T ) (cid:1) κ ( x, t ; T ) = m ( x ) − qθ ( x ) σ ( x ) − q ˆ ξ ( x, t ; T ) σ ( x ) + φ ′ ( x ) φ ( x ) (cid:0) σ ( x ) + σ ( x ) (cid:1) . (5.21) emark 5.2. A way to understand the above theorem is to consider the commutative diagram (Ω , F , ˆ P ) (Ω , F , Q ) non-ergodic HJB (optimal control ˆ ξ ( x, t ; T ) in drift of X ) (Ω , F , ˜ P ) (Ω , F , P ) ergodic HJB (optimal control ξ ∗ ( x ) in drift of X ) d ˜ P d ˆ P M T = d P d ˜ P d Q d P To be able to express the dual value function in terms of an ergodic HJB eigenpair, we have first to switch to a measure ˆ P under which the drift of the underlying diffusion factor process X is independent of time t and the time horizon T . Underthis measure the corresponding HJB equation is ergodic and we can rewrite the multiplicate functional in terms of theassociated eigenpair in the sense of Hansen–Scheinkman (even though the functions in the multiplicative functional dependon the time horizon T ). After that, we can switch back to the original, maturity-dependent drift process. This procedurecan be performed as long as all measure changes are well defined, i.e., the corresponding Radon–Nikod´ym derivatives aretrue martingales (see A8 and A10), which means that the original optimal control ˆ ξ is not “too far from ergodic” optimalcontrol ξ ∗ . The long-term asymptotic behavior of the function v ( χ, T ) is given by v ( χ, T ) ≃ e − λT φ ( χ ) , thus in the decomposition in Eq.(5.20), the expectation E Q h φ ( X T ) e R T f ( X s ,s ; T ) ds i can be understood as an error term. Our derivation of the long-term sensitivity relies mainly on estimations of this errorterm. This section studies the sensitivity of the optimal expected utility with respect to the initial factor χ = X . Using the dualformulation of Eq.(5.3), the initial-value sensitivity in Eq.(3.1) can be expressed as ∂∂χ ln (cid:12)(cid:12)(cid:12) sup Π ∈X E P (cid:2) U (Π T ) (cid:3)(cid:12)(cid:12)(cid:12) = (1 − p ) ∂∂χ ln v ( χ, T )and thus we are interested in the sensitivity ∂∂χ ln v ( χ, T ) for large time T . The sensitivity for large time T is described inTheorem 3.2, which states that ∂∂χ ln v ( χ, T ) is asymptotically equal to φ ′ ( χ ) φ ( χ ) . The proof is following.
Proof of Theorem 3.2.
The function φ is continuously differentiable by A6. From Eq.(5.20), applying the chain rule, weobtain the differentiability of v ( χ, t ) and ∂∂χ v ( χ, T ) v ( χ, T ) = φ ′ ( χ ) φ ( χ ) + ∂∂χ E Q (cid:2) φ ( X T ) e R T f ( X s ,s ; T ) ds (cid:3) E Q (cid:2) φ ( X T ) e R T f ( X s ,s ; T ) ds (cid:3) . The nominator of the second term goes to zero by assumption, and A6 and Eq.(5.20) give the convergence to a positiveconstant of the denominator. This completes the proof.In order to utilize Theorems 3.2 and 5.1, we have to provide sufficient conditions under which the mapping χ E Q (cid:2) φ ( X T ) e R T f ( X s ,s ; T ) ds (cid:3) is continuously differentiable and its derivative converges to zero as T → ∞ . To denote thedependence of the solution X of the SDE (2.2) on the initial value x , we write X x . Assume that for almost all ω ∈ Ω themap x X xt is continuously differentiable and the derivative process ( Y t ) ≤ t ≤ T := ( ∂X xt ∂x ) ≤ t ≤ T , which is called the firstvariation process, satisfies dY t = κ x ( X t , t ; T ) Y t dt + σ ′ ( X t ) Y t dB ,t + σ ′ ( X t ) Y t dB ,t , Y = 1 . (6.1)This holds, as a particular case, if the derivative of κ ( · , t ; T ) is jointly continuous in x and t for fixed T and σ and σ arecontinuously differentiable with bounded derivatives (for details, see (Protter, 2005, Theorem V.39)). Proposition 6.1.
Additionally to A1 – 10, assume that for almost all ω ∈ Ω the map x X xt is continuously differentiableand the first variation process ( Y t ) ≤ t ≤ T satisfies Eq. (6.1) and f is continuously differentiable. Suppose that there exist anopen neighborhood I χ of χ and positive constants u, v, w with u + v + w = 1 satisfying the following conditions. i) As a function of two variables ( x, T ) , the expectation Γ u ( x, T ) = E Q h φ u ( X T ) e u R T f ( X s ,s ; T ) ds (cid:12)(cid:12)(cid:12) X = x i is uniformly bounded on I χ × (0 , ∞ ) . (ii) As a function of two variables ( x, T ) , the expectation E Q (cid:12)(cid:12)(cid:12)(cid:12) φ ′ ( X T ) φ ( X T ) (cid:12)(cid:12)(cid:12)(cid:12) v is uniformly bounded on I χ × (0 , ∞ ) . (iii) As a function of two variables ( x, T ) , the expectation E Q | Y T ; T | w is uniformly bounded on I χ for each T and convergesto zero as T → ∞ for each x ∈ I χ . (iv) The expectation E Q (cid:20)(cid:18)Z T | f x ( X s , s ; T ) Y s ; T | ds (cid:19) m (cid:21) is uniformly bounded on I χ for each T and converges to zero as T → ∞ for each x ∈ I χ . Here, m = uu − , i.e., u + m = 1 . Then the map x E Q (cid:2) φ ( X T ) e R T f ( X s ,s ; T ) ds (cid:12)(cid:12) X = x (cid:3) is continuously differentiable in x on I χ , and ∂∂x E Q h φ ( X T ) e R T f ( X s ,s ; T ) ds (cid:12)(cid:12)(cid:12) X = x i converges to zero as T → ∞ . Proof.
First we observe that ∂∂x E Q (cid:16) φ ( X T ) e R T f ( X s ,s ; T ) ds (cid:17) = E Q (cid:16) φ ( X T ) e R T f ( X s ,s ; T ) ds Z T f x ( X s , s ; T ) Y s ds − φ ′ ( X T ) φ ( X T ) e R T f ( X s ,s ; T ) ds Y T (cid:17) holds and the derivative is a continuous function of x. This equality can be obtained by interchanging the derivative andthe expectation, and this is justified since E Q (cid:12)(cid:12)(cid:12)(cid:12)(cid:16) φ ( X T ) (cid:17) ′ e R T f ( X s ,s ; T ) ds Y T + 1 φ ( X T ) e R T f ( X s ,s ; T ) ds Z T f x ( X s , s ; T ) Y s ds (cid:12)(cid:12)(cid:12)(cid:12) ≤ E Q (cid:12)(cid:12)(cid:12)(cid:12) φ ′ ( X t ) φ ( X t ) e R T f ( X s ,s ; T ) ds Y T (cid:12)(cid:12) + E Q (cid:12)(cid:12)(cid:12)(cid:12) φ ( X T ) e R T f ( X s ,s ; T ) ds Z T f x ( X s , s ; T ) Y s ds (cid:12)(cid:12)(cid:12)(cid:12) ≤ Γ u ( x, T ) u (cid:18) E Q (cid:12)(cid:12)(cid:12)(cid:12) φ ′ ( X t ) φ ( X t ) (cid:12)(cid:12)(cid:12)(cid:12) v (cid:19) v ( E Q | Y T | w ) w + Γ u ( x, T ) u (cid:18) E Q (cid:16)Z T | f x ( X s , s ; T ) Y s | ds (cid:17) m (cid:19) m is uniformly bounded on I χ by (i)-(iv). Moreover, the same inequality gives that the derivative goes to zero as T → ∞ . This section studies the sensitivities with respect to the drift and volatility perturbations. The arguments in this sectionis similar to Park (2018).
We provide a precise meaning of the perturbed the drift and volatility functions.
B 1.
Let m ǫ , σ ,ǫ , σ ,ǫ , b ǫ , ς ǫ be continuous functions in the variables ( ǫ, x ) ∈ I × R for a neighborhood I of such thatthey are continuously differentiable in ǫ on I and m = m, σ , = σ , σ , = σ , b = b, ς = ς. B 2.
For each ǫ ∈ I, the functions m ǫ , σ ,ǫ , σ ,ǫ , b ǫ , ς ǫ satisfy A1 – 10. The domain ( ℓ ǫ , r e ) in A1 of the process X ǫ maydepend on ǫ, and the constant C in A6 can also depend on ǫ. rom theses assumptions, we can construct the following objects. Let X ǫ be the solution of the SDE dX ǫt = m ǫ ( X ǫt ) dt + σ ,ǫ ( X ǫt ) dW ,t + σ ,ǫ ( X ǫt ) dW ,t , X ǫ = χ with perturbation parameter ǫ. The initial value χ is not perturbed. We denote by X ǫ the family of wealth processes ofadmissible portfolios in the perturbed market model. Define v ǫ ( χ, T ) := E P (cid:2) ( ˆ Y ǫT ) q (cid:3) = E P (cid:2) ( ˆ Y ǫT ) q (cid:12)(cid:12) X = χ (cid:3) where ˆ Y ǫT is the optimizer of the dual problem in the perturbed market. We are interested in the sensitivity ∂∂ǫ (cid:12)(cid:12)(cid:12) ǫ =0 ln (cid:12)(cid:12)(cid:12) sup Π ∈X ǫ E P (cid:2) U (Π T ) (cid:3)(cid:12)(cid:12)(cid:12) for large time T. From the dual formulation in Eq.(5.3), we know that the long-term sensitivity can be obtained byevaluating ∂∂ǫ (cid:12)(cid:12)(cid:12) ǫ =0 ln v ǫ ( χ, T ) . We can transform this sensitivity into a simpler form similar to Eq.(5.20) by using an exponential change of measure.Then v ǫ ( χ, T ) = e − λ ǫ T φ ǫ ( χ ) E Q ǫ h φ ǫ ( X ǫT ) e R T f ǫ ( X ǫs ,s ; T ) ds i (7.1)and the Q ǫ -dynamics of X is dX ǫt = κ ǫ ( t, X ǫt ; T ) dt + σ ,ǫ ( X ǫt ) dB ǫ ,t + σ ,ǫ ( X ǫt ) dB ǫ ,t , ≤ t ≤ T for a two-dimensional Q ǫ -Brownian motion ( B ǫ ,t , B ǫ ,t ) t ≥ . Here, the functions f ǫ and κ ǫ are defined as f ǫ ( x, t ; T ) := − q − q ) (cid:0) ξ ∗ ǫ ( x ) − ˆ ξ ǫ ( x, t ; T ) (cid:1) κ ǫ ( x, t ; T ) := m ǫ ( x ) − qθ ǫ ( x ) σ ,ǫ ( x ) − q ˆ ξ ǫ ( x, t ; T ) σ ,ǫ ( x ) + φ ′ ǫ ( x ) φ ǫ ( x ) (cid:0) σ ,ǫ ( x ) + σ ,ǫ ( x ) (cid:1) where θ ǫ , ˆ ξ ǫ , ξ ∗ ǫ , φ ǫ , are functions defined as in Eq.(2.3), Eq.(5.7), Eq.(5.12), A6, respectively for the perturbed market.We use the prime notation to denote the derivative with respect to x. For the sensitivity analysis, we assume the following regularity conditions. We want to separate the perturbation effectsof the underlying diffusion process and the functionals applied to it. Therefore, we define w η,ǫ ( χ, T ) := E Q ǫ h φ η ( X ǫT ) e R T f η ( X ǫs ,s ; T ) ds i so that v ǫ ( χ, T ) = e − λ ǫ T φ ǫ ( χ ) w ǫ,ǫ ( χ, T ) . We call this function w the error term. Theorem 7.1.
Additionally to B1 – 2, we assume the following conditions:(i) The two functions ǫ λ ǫ and ǫ φ ǫ ( χ ) are continuously differentiable on I. (ii) The partial derivative ∂∂η w η,ǫ ( χ, T ) exists and is continuous on I . Moreover, lim T →∞ T ∂∂η (cid:12)(cid:12)(cid:12) η =0 w η, ( χ, T ) = 0 . (iii) The partial derivative ∂∂ǫ w η,ǫ ( χ, T ) exists and is continuous on I . Moreover, lim T →∞ T ∂∂ǫ (cid:12)(cid:12)(cid:12) ǫ =0 w ,ǫ ( χ, T ) = 0 . Then the perturbed function ln v ǫ ( χ, T ) is differentiable at ǫ = 0 and T ∂∂ǫ (cid:12)(cid:12)(cid:12) ǫ =0 ln v ǫ ( χ, T ) (7.2)= − ∂λ ǫ ∂ǫ (cid:12)(cid:12)(cid:12) ǫ =0 + ∂∂ǫ (cid:12)(cid:12) ǫ =0 φ ǫ ( χ ) T φ ( χ ) + ∂∂ǫ (cid:12)(cid:12) ǫ =0 E Q (cid:2) φ ǫ ( X T ) e R T f ǫ ( X s ,s ; T ) ds (cid:3) T E Q (cid:2) φ ( X T ) e R T f ( X s ,s ; T ) ds (cid:3) + ∂∂ǫ (cid:12)(cid:12) ǫ =0 E Q ǫ (cid:2) φ ( X ǫT ) e R T f ( X ǫs ,s ; T ) ds (cid:3) T E Q (cid:2) φ ( X T ) e R T f ( X s ,s ; T ) ds (cid:3) . Furthermore, lim T →∞ T ∂∂ǫ (cid:12)(cid:12)(cid:12) ǫ =0 ln v ǫ ( χ, T ) = − ∂λ ǫ ∂ǫ (cid:12)(cid:12)(cid:12) ǫ =0 . (7.3) roof. Define a function V on I by V ( ǫ , ǫ , ǫ , ǫ ) := e − λ ǫ T φ ǫ ( χ ) E Q ǫ h φ ǫ ( X ǫ T ) e R T f ǫ ( X ǫ s ,s ; T ) ds i = e − λ ǫ T φ ǫ ( χ ) w ǫ ,ǫ ( χ, T )then v ǫ ( χ, T ) = V ( ǫ, ǫ, ǫ, ǫ ) . The chain rule gives the differentiability of ln v ǫ ( χ, T ) at ǫ = 0 and allows us to write thederivative as in Eq.(7.2). Because E Q ( φ ( X T ) e R T f ( X s ,s ; T ) ds ) converges to a positive constant as T → ∞ by A6 andEq.(7.1), we obtain Eq.(7.3) from conditions (i) – (iii) and Eq.(7.2).Let us discuss conditions (i) – (iii) in Theorem 7.1 given above in more detail. Condition (i) is satisfied for manyfinancially meaningful models. Condition (ii) is easy to check because the continuous differentiability of E Q h φ ǫ ( X T ) e R T f ǫ ( X s ,s ; T ) ds i is a standard problem of differentiation and integration. An easier to check condition that is sufficient to imply condition(ii) and is used in the calculation of the examples of Section 4 will be given in Appendix C. Conditions (i) and (ii) can bechecked case-by-case, thus we do not go into further details of the first three terms of Eq.(7.2). However, condition (iii) isinvolved as it concerns the perturbation in the underlying process X ǫ and the measure Q ǫ , which are not trivial to analyze.We will provide a sufficient condition such that condition (iii) holds true in Theorems 7.3 and 7.5.For the analysis of these parameter sensitivities, the following expression for the Q ǫ -dynamics of X is useful. Let σ ǫ ( · ) := q σ ,ǫ ( · ) + σ ,ǫ ( · ) , σ ( · ) := σ ( · )and define a new process B ǫ = ( B ǫt ) t ≥ by dB ǫt = σ ,ǫ ( X ǫt ) σ ǫ ( X ǫt ) dB ǫ ,t + σ ,ǫ ( X ǫt ) σ ǫ ( X ǫt ) dB ǫ ,t , B ǫ = 0 , then B ǫ is a Q ǫ -Brownian motion as can be seen by L´evy’s characterization. The Q ǫ -dynamics of X can then be writtenas dX ǫt = κ ǫ ( X ǫt , t ; T ) dt + σ ǫ ( X ǫt ) dB ǫt . Remark 7.2.
If we consider the problem of the sensitivity of the expected utility stemming from optimizing the long termgrowth rate, i.e., ∂∂ǫ (cid:12)(cid:12)(cid:12) ǫ =0 inf Π ∈X ǫ lim T →∞ T ln (cid:12)(cid:12)(cid:12) E P (cid:2) U (Π T ) (cid:3)(cid:12)(cid:12)(cid:12) , actually all the results in Section 7 hold true, only with less assumptions. Following the discussion at the end of Section5.1, in this case the optimal value can be expressed using the function v in Eq. (5.14) only with ξ ∗ given in Eq. (5.12) insteadof ˆ ξ . In this case we are already in an ergodic regime and no additional change of measure is needed. Thus it it is sufficientto require Assumptions A1 – A7 as well as A9 for each ǫ ∈ I where the two-dimensional Brownian motion ˜ W is replacedby ˆ W . Refer to Fleming et al. (2002) for details. In this section, we conduct a sensitivity analysis with respect to the perturbations of m ǫ , b ǫ , ς ǫ , but assume that thevolatility functions σ ,ǫ = σ , σ ,ǫ = σ are not perturbed. Under the measure Q ǫ , the perturbed process X ǫ has the form dX ǫt = κ ǫ ( X ǫt , t ; T ) dt + σ ( X ǫt ) dB ǫt so that only the drift term is perturbed. Our goal is to analyze ∂∂ǫ w η,ǫ ( χ, T ) = ∂∂ǫ E Q ǫ h φ η ( X ǫT ) e R T f η ( X ǫs ,s ; T ) ds i under this drift perturbation.Assuming that κ ǫ is continuously differentiable in ǫ on I, defineˆ φ ( · ) := inf ǫ ∈ I φ ǫ ( · ) (7.4)ˆ f ( · , t ; T ) := sup ǫ ∈ I f ǫ ( · , t ; T )ˆ g ( · , t ; T ) := sup ǫ ∈ I (cid:12)(cid:12)(cid:12) σ ( x ) ∂∂ǫ κ ǫ ( · , t ; T ) (cid:12)(cid:12)(cid:12) . We consider the following boundedness assumptions; ˆ φ ( · ) > , ˆ f ( · , t ; T ) < ∞ and ˆ g ( · , t ; T ) < ∞ . If the domain in ( ℓ ǫ , r e )in B2 does not depend on ǫ, then the three functions always satisfy these boundedness condition by replacing the interval I by a smaller interval if necessary. heorem 7.3. Additionally to B1 – 2, assume that ˆ φ ( · ) > , ˆ f ( · , t ; T ) < ∞ , ˆ g ( · , t ; T ) < ∞ and that κ ǫ is continuouslydifferentiable and f ǫ is continuous in ǫ on I. Suppose the following conditions.(i) For each T ≥ , there exists a real number ǫ = ǫ ( T ) > such that E Q h e ǫ R T ˆ g ( X s ,s ; T ) ds i is finite.(ii) There exist a real number v ≥ and a function h with lim T →∞ h ( T ) = 0 such that for all T > E Q (cid:20)(cid:16)Z T ˆ g ( X s , s ; T ) ds (cid:17) v/ (cid:21) ≤ T v h ( T ) . (iii) For each T ≥ , there is a real number ǫ > such that E Q (cid:20)Z T ˆ g v + ǫ ( X s , s ; T ) ds (cid:21) is finite.(iv) The function ˆΓ u ( T ) := E Q h φ u ( X T ) e u R T ˆ f ( X s ,s ; T ) ds i is uniformly bounded in T ≥ where u = vv − , i.e., u + v = 1 , for v from (ii).Then, for given ( χ, T ) , the partial derivative ∂∂ǫ w η,ǫ ( χ, T ) = ∂∂ǫ E Q ǫ h φ η ( X ǫT ) e R T f η ( X ǫs ,s ; T ) ds i exists and is continuous in ( η, ǫ ) on I . Moreover, for given χ, T ∂∂ǫ (cid:12)(cid:12)(cid:12) ǫ =0 w ,ǫ ( χ, T ) = 1 T ∂∂ǫ (cid:12)(cid:12)(cid:12) ǫ =0 E Q ǫ h φ ( X ǫT ) e R T f ( X ǫs ,s ; T ) ds i → T → ∞ . The proof of the above theorem is similar to the proof of Proposition A.1 in Park (2018), but for the sake of completenesswe provide the proof in Appendix B.
Remark 7.4.
One can relax the assumption in the above theorem on the continuous differentiability of κ ǫ by replacing itwith local Lipschitz continuity and defining ˆ g ( · , t ; T ) := sup ǫ ∈ I (cid:12)(cid:12)(cid:12) κ ǫ ( x, t ; T ) − κ ( x, t ; T ) ǫσ ( x ) (cid:12)(cid:12)(cid:12) . As this introduces cumbersome additional notations, we do not pursue this in the current paper.
This section discusses the volatility perturbation of the factor process. Consider B1 – 2 and the perturbed process dX ǫt = κ ǫ ( X ǫt , t ; T ) dt + σ ǫ ( X ǫt ) dB ǫt , X ǫ = χ. Contrary to the previous section, we allow for an additional perturbation of the volatility of the factor process. As this isa mathematically harder problem, we will need stronger conditions.The main tool of this section is the Lamperti transformation. We assume that ( ǫ, x ) σ ǫ ( x ) is twice continuouslydifferentiable. Fix any c ∈ ( r, ℓ ) and define ℓ ǫ ( · ) := Z · c σ ǫ ( z ) dz, ℓ ( · ) := ℓ ( · ) . As σ ǫ is positive, the function ℓ ǫ is invertible. Define two functions Φ ǫ , F ǫ and a process ˇ X ǫ byΦ ǫ ( · ) = φ ǫ (cid:0) ℓ − ǫ ( · ) (cid:1) , F ǫ ( · , t ; T ) = f ǫ (cid:0) ℓ − ǫ ( · ) , t ; T (cid:1) , ˇ X ǫt := ℓ ǫ ( X ǫt ) , and let Φ := Φ , F := F and ˇ X := ˇ X . The integral begins with a fixed constant c so that the initial value ˇ X ǫ = R χc σ ǫ ( u ) du is also perturbed if χ = c. The function v ǫ ( x, T ) we want to analyze can be expressed as v ǫ ( χ, T ) = e − λ ǫ T φ ǫ ( χ ) E Q ǫ h ǫ ( ˇ X ǫT ) e R T F ǫ ( ˇ X ǫs ,s ; T ) ds i . sing the Itˆo formula, it is easy to show that the Q ǫ -dynamics of ˇ X ǫ is d ˇ X ǫt = γ ( ˇ X ǫt ) dt + dB ǫt , ˇ X ǫ = ℓ ǫ ( χ )where γ ( · ) := κ ǫ (cid:0) ℓ − ǫ ( · ) , t ; T (cid:1) σ ǫ (cid:0) ℓ − ǫ ( · ) (cid:1) − σ ′ ǫ (cid:0) ℓ − ǫ ( · ) (cid:1) . Let U be an open neighborhood of ℓ ( χ ) and define˜ w η,ǫ (ˇ x, T ) := E Q ǫ h η ( ˇ X ǫT ) e R T F η ( ˇ X ǫs ,s ; T ) ds (cid:12)(cid:12)(cid:12) ˇ X ǫ = ˇ x i for ( η, ǫ, ˇ x, T ) ∈ I × I × U × [0 , ∞ ) so that v ǫ ( χ, T ) = e − λ ǫ T φ ǫ ( χ ) ˜ w ǫ,ǫ (cid:0) ℓ ǫ ( χ ) , T (cid:1) . Under these circumstances, we obtain the following theorem. The proof is similar to that of Theorem 7.2.
Theorem 7.5.
Additionally to B1 – 2, assume that ( ǫ, x ) σ ǫ ( x ) is twice continuously differentiable. Suppose condition(i) in Theorem 7.1 and the following conditions.(i) The partial derivative ∂∂ ˇ x ˜ w η,ǫ (ˇ x, T ) exists and is continuous in ( η, ǫ, ˇ x ) on I × I × U. Moreover, lim T →∞ T ∂∂ ˇ x (cid:12)(cid:12)(cid:12) ˇ x = ℓ ( χ ) ˜ w , (ˇ x, T ) = 0 . (ii) The partial derivative ∂∂η ˜ w η,ǫ (ˇ x, T ) exists and is continuous in ( η, ǫ, ˇ x ) on I × I × U. Moreover, lim T →∞ T ∂∂η (cid:12)(cid:12)(cid:12) η =0 ˜ w η, (cid:0) ℓ ( χ ) , T (cid:1) = 0 . (iii) The partial derivative ∂∂ǫ ˜ w η,ǫ (ˇ x, T ) exists and is continuous in ( η, ǫ, ˇ x ) on I × I × U. Moreover, lim T →∞ T ∂∂ǫ (cid:12)(cid:12)(cid:12) ǫ =0 ˜ w ,ǫ (cid:0) ℓ ( χ ) , T (cid:1) = 0 . Then ˜ w η,ǫ ( x, T ) (thus ln v ǫ ( x, T ) ) is differentiable in ǫ on I and ∂∂ǫ (cid:12)(cid:12)(cid:12) ǫ =0 w ǫ,ǫ ( χ, T ) = ∂∂ǫ (cid:12)(cid:12)(cid:12) ǫ =0 ˜ w ǫ,ǫ (cid:0) ℓ ǫ ( χ ) , T (cid:1) = ∂∂ǫ (cid:12)(cid:12)(cid:12) ǫ =0 ℓ ǫ ( χ ) · ∂∂ ˇ x (cid:12)(cid:12)(cid:12) ˇ x = ℓ ( χ ) ˜ w , (ˇ x, T ) + ∂∂η (cid:12)(cid:12)(cid:12) η =0 ˜ w η, (cid:0) ℓ ( χ ) , T (cid:1) + ∂∂ǫ (cid:12)(cid:12)(cid:12) ǫ =0 ˜ w ,ǫ (cid:0) ℓ ( χ ) , T (cid:1) . (7.5) Finally, lim T →∞ T ∂∂ǫ (cid:12)(cid:12)(cid:12) ǫ =0 ln v ǫ ( χ, T ) = − ∂λ ǫ ∂ǫ (cid:12)(cid:12)(cid:12) ǫ =0 . This theorem has an important implication, namely that the volatility sensitivity of the error term w is a sum of theinitial value sensitivity, the functional sensitivity and the drift sensitivity of the error term. Condition (ii) in the abovetheorem is about the sensitivity with respect to the functional perturbation, which is corresponding to condition (ii) inTheorem 7.1. Condition (iii) in the above theorem is about the sensitivity with respect to the drift corresponding tocondition (iii) in Theorem 7.1, which can be analyzed in the same way in Section 7.2. In the special case c = χ we canomit condition (i) in the above theorem since the initial value is not perturbed. Moreover, Eq.(7.5) can be written as ∂∂ǫ (cid:12)(cid:12)(cid:12) ǫ =0 w ǫ,ǫ ( χ, T ) = ∂∂ǫ (cid:12)(cid:12)(cid:12) ǫ =0 ˜ w ǫ,ǫ ( T ) = ∂∂η (cid:12)(cid:12)(cid:12) η =0 ˜ w η, ( T ) + ∂∂ǫ (cid:12)(cid:12)(cid:12) ǫ =0 ˜ w ,ǫ ( T ) . In this paper, we conducted a sensitivity analysis of the long-term expected utility of optimal portfolios in an incompletemarket given by a factor model. The main purpose was to find the long-term sensitivity, that is, the extent how much theoptimal expected utility is affected in the long run for small changes of the underlying factor model. We calculated twokinds of sensitivities; The first is the initial factor sensitivity. For the initial value χ = X of the factor process, we studythe behavior of ∂∂χ sup Π ∈X E P (cid:2) U (Π T ) (cid:3) or large T. The second kind is the drift and volatility sensitivities. For a perturbation parameter ǫ, consider a perturbedasset price S ǫ with S = S and the family X ǫ of wealth processes of admissible portfolios with the perturbed asset model S ǫ . For the long-term sensitivity, we are interested in the behavior of ∂∂ǫ (cid:12)(cid:12)(cid:12) ǫ =0 sup Π ∈X ǫ E P (cid:2) U (Π T ) (cid:3) for large T. To achieve this, we employed several techniques. The primal utility maximization problem was transformed into thedual problem. Then, we approximated the solution of the dual problem by an HJB equation. The long-term behavior ofthe optimal expected utility can be characterized by a solution pair ( λ, φ ) of the corresponding ergodic HJB equation, andwe demonstrated that this solution pair determines the long-term sensitivities. The solution v of the dual problem can bedecomposed as v ( χ, T ) = e − λT φ ( χ ) E Q h φ ( X T ) e R T f ( X s ,s ; T ) ds i . We regarded the expectation in this expression as an error term and then found sufficient conditions under which this errorterm is negligible. We provided examples of explicit results for several market models such as the Kim–Omberg model forstochastic excess returns and the Heston stochastic volatility model.
Acknowledgement.
Hyungbin Park was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government(MSIT) (No. 2018R1C1B5085491 and No. 2017R1A5A1015626).
References
J Backhoff and F J Silva. Sensitivity results in stochastic optimal control: A Lagrangian perspective.
ESAIM. Control,Optimisation and Calculus of Variations , 23(1):39–70, 2017.Julio Backhoff Veraguas and Francisco J Silva. Sensitivity analysis for expected utility maximization in incomplete Brownianmarket models.
Mathematics and Financial Economics , 12(3):387–411, 2018.Anna Battauz, Marzia De Donno, and Alessandro Sbuelz. Kim and Omberg revisited: the duality approach.
Journal ofProbability and Statistics , 2015, 2015.Freddy Delbaen and Walter Schachermayer. A general version of the fundamental theorem of asset pricing.
MathematischeAnnalen , 300(1):463–520, 1994.Steffen Dereich, Andreas Neuenkirch, and Lukasz Szpruch. An Euler-type method for the strong approximation of theCox–Ingersoll–Ross process.
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences , 468(2140):1105–1115, 2011.Wendell H Fleming and William M McEneaney. Risk-sensitive control on an infinite time horizon.
SIAM Journal onControl and Optimization , 33(6):1881–1915, 1995.WH Fleming, SJ Sheu, et al. Risk-sensitive control and an optimal investment model ii.
The Annals of Applied Probability ,12(2):730–767, 2002.Martin Forde and Antoine Jacquier. The large-maturity smile for the Heston model.
Finance and Stochastics , 15(4):755–780, 2011.Paolo Guasoni and Scott Robertson. Portfolios and risk premia for the long run.
The Annals of Applied Probability , 22(1):239–284, 2012.Lars Peter Hansen and Jos´e A Scheinkman. Long-term risk: An operator approach.
Econometrica , 77(1):177–234, 2009.Daniel Hern´andez-Hern´andez and Alexander Schied. Robust utility maximization in a stochastic factor model.
Statistics& Decisions. International Journal for Statistical Theory and Related Fields , 24(1):109–125, 2006.Steven L Heston. A closed-form solution for options with stochastic volatility with applications to bond and currencyoptions.
The Review of Financial Studies , 6(2):327–343, 1993.Hidehiro Kaise and Shuenn-Jyi Sheu. Ergodic type Bellman equations of first order with quadratic Hamiltonian.
AppliedMathematics and Optimization , 59(1):37–73, 2009.Ioannis Karatzas and Steven E Shreve. Brownian motion. In
Brownian Motion and Stochastic Calculus , pages 47–127.Springer, 1998. artin Keller-Ressel. Moment explosions and long-term behavior of affine stochastic volatility models. MathematicalFinance: An International Journal of Mathematics, Statistics and Financial Economics , 21(1):73–98, 2011.Tong Suk Kim and Edward Omberg. Dynamic nonmyopic portfolio behavior.
The Review of Financial Studies , 9(1):141–161, 1996.Fima Klebaner and Robert Liptser. When a stochastic exponential is a true martingale. Extension of the Beneˇs method.
Theory of Probability & Its Applications , 58(1):38–62, 2014.Thomas Knispel. Asymptotics of robust utility maximization.
The Annals of Applied Probability , 22(1):172–212, 2012.D Kramkov and W Schachermayer. The asymptotic elasticity of utility functions and optimal investment in incompletemarkets.
The Annals of Applied Probability , 9(3):904–950, 1999.Dmitry Kramkov and Mihai Sˆırbu. Sensitivity analysis of utility-based prices and risk-tolerance wealth processes.
TheAnnals of Applied Probability , 16(4):2140–2194, 2006.Kasper Larsen and Gordan ˇZitkovi´c. Stability of utility-maximization in incomplete markets.
Stochastic Processes andtheir Applications , 117(11):1642–1662, 2007.Kasper Larsen, Oleksii Mostovyi, and Gordan ˇZitkovi´c. An expansion in the model space in the context of utility maxi-mization.
Finance and Stochastics , 22(2):297–326, 2018.Ren Liu and Johannes Muhle-Karbe. Portfolio choice with stochastic investment opportunities: a user’s guide. arXivpreprint arXiv:1311.1715 , 2013.Phillip Monin and Thaleia Zariphopoulou. On the optimal wealth process in a log-normal market: Applications to riskmanagement.
Journal of Financial Engineering , 1(2), 2014.Oleksii Mostovyi. Asymptotic analysis of the expected utility maximization problem with respect to perturbations of thenum´eraire. arXiv preprint arXiv:1805.11427 , 2018.Oleksii Mostovyi and Mihai Sˆırbu. Sensitivity analysis of the utility maximization problem with respect to model pertur-bations. arXiv preprint arXiv:1705.08291 , 2017.Hyungbin Park. Sensitivity analysis of long-term cash flows.
Finance and Stochastics , 22(4):773–825, 2018.Philip E. Protter.
Stochastic integration and differential equations , volume 21 of
Stochastic Modelling and Applied Proba-bility . Springer-Verlag, Berlin, 2005.Likuan Qin and Vadim Linetsky. Positive eigenfunctions of Markovian pricing operators: Hansen–Scheinkman factorization,Ross recovery, and long-term pricing.
Operations Research , 64(1):99–117, 2016.Scott Robertson and Hao Xing. Large time behavior of solutions to semilinear equations with quadratic growth in thegradient.
SIAM Journal on Control and Optimization , 53(1):185–212, 2015.
A Motivation for the ergodic HJB equation
In this section, we derive the ergodic HJB equation and provide the motivation of A4 – 7. These assumptions originate from the dynamicprogramming principle. Let M be the set of all progressively measurable processes ξ such that R t ξ s ds < ∞ a.s. for each t. Then v ( x, T ) = sup Y ∈Y E P (cid:2) Y qT (cid:3) = sup ξ ∈M E P h e − q R T θ ( Xs ) dW ,s − q R T θ Xs ) ds − q R T ξs dW ,s − q R T ξ s ds i = sup ξ ∈M E ˆ P h e q q − R T θ Xs )+ ξ s ) ds i where d ˆ P d P (cid:12)(cid:12)(cid:12) F T = E (cid:16) − q Z · θ ( X s ) dW ,s − q Z · ξ s dW ,s (cid:17) T defines a martingale due to A7. The ˆ P -dynamics of X is dX t = ( m ( X t ) − qθ ( X t ) σ ( X t ) − qξ t σ ( X t )) dt + σ ( X ) d ˆ W ,t + σ ( X t ) d ˆ W ,t for a ˆ P -Brownian motion ( ˆ W ,t , ˆ W ,t ) . We regard the process X as a state variable and ξ as a control variable. The standard argument of thedynamic programming principle says that the value function u ( x, t ) := sup ξ ∈M E ˆ P Xt = x h e R Tt l ( ξs,Xs ) ds i satisfies u t + 12 ( σ ( x ) + σ ( x )) u xx + sup ξ ∈ R { h ( ξ, x ) u x + l ( ξ, x ) u } = 0 , u ( x, T ) = 1 . (A.1)The optimal control of Eq.(A.1) is given by ˆ ξ ( x, t ; T ) = − σ ( x ) u x ( x, t )(1 − q ) u ( x, t ) .
21t is convenient to consider an initial condition at time 0 ,v ( x, t ) = sup ξ ∈M E ˆ P X x h e R t l ( ξs,Xs ) ds i . We know that from the Markov property v ( x, t ) = sup ξ ∈M E ˆ P X x h e R t l ( ξs,Xs ) ds i = sup ξ ∈M E ˆ P XT − t = x h e R TT − t l ( ξs,Xs ) ds i = u ( x, T − t ) . The function v ( x, t ) satisfies v t = 12 ( σ ( x ) + σ ( x )) v xx + sup ζ ∈ R { l ( ζ, x ) v + h ( ζ, x ) v x } , v (0 , x ) = 1 . (A.2)The optimal control of Eq.(A.2) is given by ˆ ζ ( x, t ; T ) = − σ ( x ) v x ( x, t )(1 − q ) v ( x, t )and it is clear that ˆ ξ ( x, t ; T ) = ˆ ζ ( x, T − t ; T ) = − σ ( x ) v x ( x, T − t )(1 − q ) v ( x, T − t ) , which motivates Assumption 5 and Eq.(5.7).The ergodic HJB equation is useful to obtain the growth rate − λ and to understand the behavior of the optimal function ˆ ξ. Heuristically,by taking v ( t, x ) = e − λt φ ( x ) in Eq.(5.4), we have − λφ ( x ) = 12 ( σ ( x ) + σ ( x )) φ xx + sup ζ ∈ R { l ( ζ, x ) φ + h ( ζ, x ) φ x } . This is a kind of an eigenvalue/eigenfunction problem. The unknown is a pair ( λ, φ ) and the solution pair is not unique in general.A6 assumesthat a specific solution pair ( λ, φ ) of this ergodic HJB equation approximates the function v defined in Eq.(5.2), which is also a solution ofthe original HJB equation (5.4). Many authors discuss sufficient conditions for this assumption. Refer to Assumption 4.1 in Knispel (2012)and Theorem 3.3 in Fleming and McEneaney (1995). B Proof of Theorem 7.3
Proof of Theorem 7.3 relies on the following proposition, whose proof is rather long and tedious. We recall the functions ˆ φ, ˆ f and ˆ g defined inEq.(7.4). The proof of this proposition is similar to the proof of Proposition A.1 in Park (2018), but for the sake of completeness we providethe proof here. Proposition B.1.
Additionally to B1 – 2, assume that ˆ φ ( · ) > , ˆ f ( · , t ; T ) < ∞ , ˆ g ( · , t ; T ) < ∞ and that κ ǫ is continuously differentiableand f ǫ is continuous in ǫ on I. Fix
T > and suppose the following conditions.(i) There exists a real number ǫ > such that E Q h e ǫ R T g Xs,s ; T ) ds i is finite.(ii) There exist real numbers v ≥ and ǫ > such that E Q Z T ˆ g v + ǫ ( X s , s ; T ) ds is finite.(iii) The function ˆΓ u ( T ) := E Q (cid:20) φ u ( X T ) e u R T f ( Xs,s ; T ) ds (cid:21) is finite where u = vv − , i.e., u + v = 1 , for v from (ii).Then, for given ( χ, T ) , the partial derivative ∂∂ǫ w η,ǫ ( χ, T ) exists and ∂∂ǫ w η,ǫ ( χ, T ) = ∂∂ǫ E Q ǫ (cid:20) φ η ( X ǫT ) e R T fη ( Xǫs,s ; T ) ds (cid:21) = E Q ǫ (cid:20) φ η ( X ǫT ) e R T fη ( Xǫs,s ; T ) ds Z T ℓ ǫ ( X ǫs , s ; T ) dB ǫs (cid:21) (B.1) where ℓ ǫ ( x, t ; T ) := 1 σ ( x ) ∂∂ǫ κ ǫ ( x, t ; T ) . Moreover, the derivative is continuous in ( η, ǫ ) on I for given ( χ, T ) . Proof.
As the proof of this proposition is rather intricate, we split up in several steps. We denote ℓ ( x, t ; T ) := ℓ ( x, t ; T ) . (I) We prove Eq.(B.1) for ǫ = 0 , that is, ∂∂ǫ (cid:12)(cid:12)(cid:12) ǫ =0 E Q ǫ (cid:20) φ η ( X ǫT ) e R T fη ( Xǫs,s ; T ) ds (cid:21) = E Q (cid:20) φ η ( X T ) e R T fη ( Xs,s ; T ) ds Z T ℓ ( X s , s ; T ) dB s (cid:21) (B.2)This equality will be proven by the following 4 sub-steps. 22a) First, we show that ∂∂ǫ (cid:12)(cid:12)(cid:12) ǫ =0 E Q ǫ (cid:20) φ η ( X ǫT ) e R T fη ( Xǫs,s ; T ) ds (cid:21) = lim ǫ → E Q (cid:20) φ η ( X T ) e R T fη ( Xs.s ; T ) ds Z T Z ǫs ℓ ǫ ( X s , s ; T ) dB s (cid:21) for a function ℓ ǫ and a positive martingale Z ǫ defined below.(b) We prove that the integral R T ( ℓ ǫ ( X s , s ; T ) − ℓ ( X s , s ; T )) dB s goes to zero in L v as ǫ → . (c) We prove that the integral R T ( Z ǫs − ℓ ǫ ( X s , s ; T ) dB s goes to zero in L v as ǫ → . (d) We show that steps (b) and (c) implylim ǫ → E Q (cid:20) φ η ( X T ) e R T fη ( Xs,s ; T ) ds Z T ( Z ǫs ℓ ǫ ( X s , s ; T ) − ℓ ( X s , s ; T )) dB s (cid:21) = 0 , which gives Eq.(B.2).(II) Using the result of step (I), we prove Eq.(B.1) for arbitrary ǫ ∈ I. (III) We prove that the derivative is continuous on I , which can be obtained by showing H ǫT converges to H T in L v as ǫ → H ǫT and H T are defined in Eq.(B.7). We conduct the following sub-steps.(a) First, show that ǫ Z T ( ℓ ǫ ℓ ǫ )( X s , s ; T ) ds · Z ǫT → L v as ǫ → . (b) We prove that Z T ℓ ǫ ( X s , s ; T ) dB s · Z ǫT → Z T ℓ ( X s , s ; T ) dB s in L v as ǫ → . Step (I) – (a). We first show Eq.(B.1) at ǫ = 0 . Define a function ℓ ǫ ( x, t ; T ) by ℓ ǫ ( x, t ; T ) = κǫ ( x,t ; T ) − κ ( x,t ; T ) ǫσ ( x ) if ǫ = 0 , σ ( x ) ∂∂ǫ (cid:12)(cid:12)(cid:12) ǫ =0 κ ǫ ( x, t ; T ) if ǫ = 0 , so that κ ǫ ( x, t ; T ) = κ ( x, t ; T ) + ǫℓ ǫ ( x, t ; T ) σ ( x ) . From the definition of ℓ ǫ ( x, t ; T ) , it is clear that ℓ ( x, t ; T ) = ℓ ( x, t ; T ) = ℓ ( x, t ; T ) . By the mean-value theorem, we have that | ℓ ǫ ( x, t ; T ) | ≤ ˆ g ( x, t ; T ) . For | ǫ | ≤ ǫ / , define Z ǫT := d Q ǫ d Q = E (cid:18) ǫ Z · ℓ ǫ ( X t , t ; T ) dB t (cid:19) T , then this local martingale process ( Z ǫt ) ≤ t ≤ T is a martingale since the Novikov condition is satisfied by condition (i). We then have that E Q ǫ (cid:20) φ η ( X ǫT ) e R T fη ( Xǫs,s ; T ) ds (cid:21) = E Q (cid:20) φ η ( X T ) e R T fη ( Xs,s ; T ) ds Z ǫT (cid:21) . From the equality Z ǫT − ǫ = Z T Z ǫs ℓ ǫ ( X s , s ; T ) dB s derived by the Itˆo formula, it follows that ∂∂ǫ (cid:12)(cid:12)(cid:12) ǫ =0 E Q ǫ (cid:20) φ η ( X ǫT ) e R T fη ( Xǫs,s ; T ) ds (cid:21) = ∂∂ǫ (cid:12)(cid:12)(cid:12) ǫ =0 E Q (cid:20) φ η ( X T ) e R T fη ( Xs,s ; T ) ds Z ǫT (cid:21) = lim ǫ → E Q (cid:20) φ η ( X T ) e R T fη ( Xs,s ; T ) ds Z ǫT − ǫ (cid:21) = lim ǫ → E Q (cid:20) φ η ( X T ) e R T fη ( Xs.s ; T ) ds Z T Z ǫs ℓ ǫ ( X s , s ; T ) dB s (cid:21) . (B.3) Step (I) – (b). We show that the integral R T ( ℓ ǫ ( X s , s ; T ) − ℓ ( X s , s ; T )) dB s goes to zero in L v as ǫ → . By the Burkholder–Davis–Gundyinequality and the Jensen inequality, E Q (cid:12)(cid:12)(cid:12)(cid:12)Z T ( ℓ ǫ ( X s , s ; T ) − ℓ ( X s , s ; T )) dB s (cid:12)(cid:12)(cid:12)(cid:12) v ≤ c v E Q (cid:12)(cid:12)(cid:12)(cid:12)Z T ( ℓ ǫ ( X s , s ; T ) − ℓ ( X s , s ; T )) ds (cid:12)(cid:12)(cid:12)(cid:12) v/ ≤ c v T v − E Q Z T | ℓ ǫ ( X s , s ; T ) − ℓ ( X s , s ; T ) | v ds for some positive constant c v in the Burkholder–Davis–Gundy inequality. Because | ℓ ǫ − ℓ | v ≤ v (cid:0) | ℓ ǫ | v + | ℓ | v (cid:1) ≤ v +1 ˆ g v and condition (ii)holds, we can apply the Lebesgue dominated convergence theorem, which implies that Z T (cid:0) ℓ ǫ ( X s , s ; T ) − ℓ ( X s , s ; T ) (cid:1) dB s converges to zero in L v as ǫ → . Step (I) – (c). We now show that Z T ( Z ǫs − ℓ ǫ ( X s , s ; T ) dB s converges to zero in L v as ǫ → . Choose a sufficiently large positive number m such that1 m + 11 + ǫ v < mv is a positive integer where ǫ is given by condition (ii). Remember that v ≥ . It follows again that E Q (cid:12)(cid:12)(cid:12)(cid:12)Z T ( Z ǫs − ℓ ǫ ( X s , s ; T ) dB s (cid:12)(cid:12)(cid:12)(cid:12) v ≤ c v E Q (cid:12)(cid:12)(cid:12)(cid:12)Z T | Z ǫs − | | ℓ ǫ | ( X s , s ; T ) ds (cid:12)(cid:12)(cid:12)(cid:12) v/ ≤ c v T v − E Q Z T | Z ǫs − | v | ℓ ǫ | v ( X s , s ; T ) ds ≤ c v T v − (cid:18) E Q Z T | Z ǫs − | mv ds (cid:19) m (cid:18) E Q Z T | ℓ ǫ | v + ǫ ( X s , s ; T ) ds (cid:19) ǫ v ≤ c v T v − (cid:18) E Q Z T | Z ǫs − | mv ds (cid:19) m (cid:18) E Q Z T ˆ g v + ǫ ( X s , s ; T ) ds (cid:19) ǫ v . The second term is finite by condition (ii).We now prove that the first expectation converges to zero as ǫ → . Consider( Z ǫt − mv = mv X i =0 (cid:16) mvi (cid:17) ( − mv − i ( Z ǫt ) i . (B.4)It is enough to show that E Q R T ( Z ǫt ) i dt converges to T as ǫ → i = 1 , , · · · , mv, because E Q Z T ( Z ǫs − mv ds = mv X i =0 (cid:16) mvi (cid:17) ( − mv − i E Q Z T ( Z ǫs ) i dt −→ T mv X i =0 (cid:16) mvi (cid:17) ( − mv − i = 0 . To show this, we apply the Lebesgue dominated convergence theorem to E Q R T ( Z ǫt ) i dt = R T E Q (cid:2) ( Z ǫt ) i (cid:3) dt : we prove that E Q (cid:2) ( Z ǫt ) i (cid:3) isuniformly bounded for small ǫ and 0 ≤ t ≤ T and that E Q (cid:2) ( Z ǫt ) i (cid:3) converges to 1 as ǫ goes to zero for fixed t. Observe that E Q (cid:2) ( Z ǫt ) i (cid:3) = E Q exp (cid:16) iǫ Z t ℓ ǫ ( X s ) dB s − iǫ Z t | ℓ ǫ | ( X s ) ds (cid:17) = E Q exp (cid:16) iǫ Z t ℓ ǫ ( X s ) dB s − i ǫ Z t | ℓ ǫ | ( X s ) ds (cid:17) · exp (cid:16) i ( i − / ǫ Z t | ℓ ǫ | ( X s ) ds (cid:17) ≤ (cid:18) E Q exp (cid:16) iǫ Z t ℓ ǫ ( X s ) dB s − i ǫ Z t | ℓ ǫ | ( X s ) ds (cid:17)(cid:19) · (cid:18) E Q exp (cid:16) i (2 i − ǫ Z t | ℓ ǫ | ( X s ) ds (cid:17)(cid:19) ≤ (cid:18) E Q exp (cid:16) i (2 i − ǫ Z t | ℓ ǫ | ( X s ) ds (cid:17)(cid:19) ≤ (cid:18) E Q exp (cid:16) i (2 i − ǫ Z t ˆ g ( X s ) ds (cid:17)(cid:19) ≤ (cid:18) E Q exp (cid:16) ǫ Z T ˆ g ( X s ) ds (cid:17)(cid:19) , (B.5)which is finite by assumption (i) for small ǫ. Here, for the second inequality, we used that the positive local martingaleexp (cid:18) iǫ Z t ℓ ǫ ( X s ) dB s − i ǫ Z t | ℓ ǫ | ( X s ) ds (cid:19) ≤ t ≤ T is a supermartingale. Thus, for small ǫ and 0 ≤ t ≤ T, the term E Q (cid:2) ( Z ǫt ) i (cid:3) is uniformly bounded by ( E Q exp (cid:0) ǫ R T ˆ g ( X s ) ds ) (cid:1) . Now we prove that E Q (cid:2) ( Z ǫt ) i (cid:3) converges to 1 as ǫ goes to zero for fixed t. We will apply the Lebesgue dominated convergent theorem toexp (cid:16) i (2 i − ǫ Z t ˆ g ( X s ) ds (cid:17) as ǫ goes to zero. Using the last inequality in Eq.(B.5), this is dominated byexp (cid:16) ǫ Z t ˆ g ( X s ) ds (cid:17) , whose expectation is finite, thus we know that E Q exp (cid:16) i (2 i − ǫ Z t ˆ g ( X s ) ds (cid:17) converges to 1 as ǫ goes to zero.1 = E Q h lim inf ǫ → ( Z ǫt ) i i ≤ lim inf ǫ → E Q (cid:2) ( Z ǫt ) i (cid:3) ≤ lim sup ǫ → E Q (cid:2) ( Z ǫt ) i (cid:3) ≤ lim ǫ → E Q exp (cid:16) i (2 i − ǫ Z t ˆ g ( X s ) ds (cid:17) = 1 . (B.6)This gives the desired result. Step (I) – (d). From Eq.(B.3), in order to show Eq.(B.2), it suffices to prove thatlim ǫ → E Q (cid:20) φ η ( X T ) e R T fη ( Xs,s ; T ) ds Z T ( Z ǫs ℓ ǫ ( X s , s ; T ) − ℓ ( X s , s ; T )) dB s (cid:21) = 0 . From the condition (iii) that ˆΓ u ( T ) = E Q (cid:20) φ u ( X T ) e u R T f ( Xs,s ; T ) ds (cid:21) is finite for u with 1 /u + 1 /v = 1, by the H¨older inequality, it is enough to show Z T ( Z ǫs ℓ ǫ ( X s , s ; T ) − ℓ ( X s , s ; T )) dB s → L v as ǫ → . Observe that Z T ( Z ǫs ℓ ǫ ( X s , s ; T ) − ℓ ( X s , s ; T )) dB s = Z T ( Z ǫs − ℓ ǫ ( X s , s ; T ) dB s + Z T ( ℓ ǫ ( X s , s ; T ) − ℓ ( X s , s ; T )) dB s . Steps (b) and (c) above imply that the two terms on the right-hand side converge to zero as ǫ → . Step (II). We now prove Eq.(B.1) for any ǫ ∈ I. Fix ǫ ∈ I and choose a small open interval J so that ǫ + J ⊆ I. We introduce another variable h to rewrite the derivative ∂∂ǫ E Q ǫ (cid:20) φ η ( X ǫT ) e R T fη ( Xǫs,s ; T ) ds (cid:21) = ∂∂h (cid:12)(cid:12)(cid:12) h =0 E Q ǫ + h (cid:20) φ η ( X ǫ + hT ) e R T fη ( Xǫ + hs ,s ; T ) ds (cid:21) . We can regard h as a perturbation parameter. It is easy to show that the perturbed functions m ǫ + h , σ ,ǫ + h , σ ,ǫ + h , b ǫ + h , v ǫ + h withperturbation parameter h ∈ J satisfy the hypothesis of this proposition. For example,sup h ∈ J (cid:12)(cid:12)(cid:12) σ ( x ) · ∂κ ǫ + h ( x ) ∂h (cid:12)(cid:12)(cid:12) ≤ sup ǫ ∈ I (cid:12)(cid:12)(cid:12) σ ( x ) · ∂κ ǫ ( x ) ∂ǫ (cid:12)(cid:12)(cid:12) ≤ ˆ g ( x ) . Thus, by applying step (I) to the perturbation parameter h , we have ∂∂h (cid:12)(cid:12)(cid:12) h =0 E Q ǫ + h (cid:20) φ η ( X ǫ + hT ) e R T fη ( Xǫ + hs ,s ; T ) ds (cid:21) = E Q ǫ (cid:20) φ η ( X ǫT ) e R T fη ( Xǫs,s ; T ) ds Z T ℓ ǫ ( X ǫs , s ; T ) dB ǫs (cid:21) , where ℓ ǫ ( x, t ; T ) = 1 σ ( x ) ∂∂h (cid:12)(cid:12)(cid:12) h =0 κ ǫ + h ( x, t ; T ) = 1 σ ( x ) ∂∂ǫ κ ǫ ( x, t ; T ) . This gives Eq.(B.1) for any ǫ ∈ I. Step (III). We show that the derivative ∂∂ǫ E Q ǫ (cid:20) φ η ( X ǫT ) e R T fη ( Xǫs,s ; T ) ds (cid:21) is jointly continuous in ( η, ǫ ) on I . Using the same argument as in Step (II), it suffices to show the continuity at ( η, ǫ ) = (0 , . We knowthat ∂∂ǫ E Q ǫ (cid:20) φ η ( X ǫT ) e R T fη ( Xǫs,s ; T ) ds (cid:21) = E Q ǫ (cid:20) φ η ( X ǫT ) e R T fη ( Xǫs,s ; T ) ds Z T ℓ ǫ ( X ǫs , s ; T ) dB ǫs (cid:21) = E Q (cid:20) φ η ( X T ) e R T fη ( Xs,s ; T ) ds (cid:16)Z T ℓ ǫ ( X s , s ; T ) dB s − ǫ Z T ( ℓ ǫ ℓ ǫ )( X s , s ; T ) ds (cid:17) Z ǫT (cid:21) For convenience, we define H ǫT := (cid:16)Z T ℓ ǫ ( X s , s ; T ) dB s − ǫ Z T ( ℓ ǫ ℓ ǫ )( X s , s ; T ) ds (cid:17) Z ǫT ; H T := H T . (B.7)Thus we want to prove that as ( η, ǫ ) → (0 , , E Q (cid:20) φ η ( X T ) e R T fη ( Xs,s ; T ) ds H ǫT (cid:21) → E Q (cid:20) φ ( X T ) e R T f ( Xs,s ; T ) ds H T (cid:21) . Condition (iii) implies by the Lebesgue dominated convergence theorem thanks to the uniform boundedness of 1 /φ η and f η over η ∈ I that1 φ η ( X T ) e R T fη ( Xs,s ; T ) ds → φ ( X T ) e R T f ( Xs,s ; T ) ds in L u as η → . It suffices to prove that H ǫT converges to H T in L v as ǫ → . This can be achieved by the following two steps.
Step (III) – (a). We show that ǫ Z T ( ℓ ǫ ℓ ǫ )( X s , s ; T ) ds · Z ǫT → L v as ǫ → . This is obtained from E Q (cid:12)(cid:12)(cid:12)(cid:12)Z T ( ℓ ǫ ℓ ǫ )( X s , s ; T ) ds · Z ǫT (cid:12)(cid:12)(cid:12)(cid:12) v ≤ E Q (cid:20)(cid:16)Z T ˆ g ( X s , s ; T ) ds (cid:17) v · ( Z ǫT ) v (cid:21) ≤ (cid:18) E Q (cid:12)(cid:12)(cid:12)Z T ˆ g ( X s , s ; T ) ds (cid:12)(cid:12)(cid:12) v (cid:19) / (cid:16) E Q (cid:2) ( Z ǫT ) v (cid:3)(cid:17) / . The expectation E Q (cid:12)(cid:12)R T ˆ g ( X s , s ; T ) ds (cid:12)(cid:12) v on the right-hand side is finite from condition (ii) and the expectation E Q (cid:2) ( Z ǫT ) v (cid:3) is uniformlybounded on I by the constant (cid:0) E Q exp( ǫ R T g ( X s ) ds ) (cid:1) using the same argument we used to derive Eq.(B.5). Step (III) – (b). We prove that Z T ℓ ǫ ( X s , s ; T ) dB s · Z ǫT → Z T ℓ ( X s , s ; T ) dB s in L v as ǫ → . Choose a sufficiently large positive number m such that1 m + 11 + ǫ v < mv is a positive integer where ǫ is given by condition (ii). It is enough to show that as ǫ → Z T ℓ ǫ ( X s , s ; T ) dB s → Z T ℓ ( X s , s ; T ) dB s in L v + ǫ (B.8)and Z ǫT → L mv . (B.9)Eq.(B.8) is obtained from condition (ii). Eq.(B.9) is from Eq.(B.4) and the fact that lim ǫ → E Q [( Z ǫt ) i ] = 1 for 0 ≤ i ≤ mv shown inEq.(B.6). 25e now shift our attention to Theorem 7.3. The proof is as follows. Proof of Theorem 7.3.
By Proposition B.1, it suffices to show thatlim T →∞ T E Q (cid:20) φ ( X T ) e R T f ( Xs,s ; T ) ds Z T ℓ ( X s , s ; T ) dB s (cid:21) = 0 . By the H¨older inequality, the Burkholder-Davis-Gundy inequality and the Jensen inequality, we know that1 T E Q (cid:12)(cid:12)(cid:12) φ ( X T ) e R T f ( Xs,s ; T ) ds Z T ℓ ( X s , s ; T ) dB s (cid:12)(cid:12)(cid:12) ≤ T ˆΓ u ( T ) u (cid:16) E Q (cid:12)(cid:12)(cid:12)Z T ℓ ( X s , s ; T ) dB s (cid:12)(cid:12)(cid:12) v (cid:17) v ≤ c ′ T ˆΓ u ( T ) u (cid:16) E Q (cid:16)Z T ℓ ( X s , s ; T ) ds (cid:17) v (cid:17) v ≤ c ′ T ˆΓ u ( T ) u (cid:16) E Q (cid:16)Z T ˆ g ( X s , s ; T ) ds (cid:17) v (cid:17) v ≤ c ′ ˆΓ u ( T ) u h ( T ) v for the positive constant c ′ in the Burkholder-Davis-Gundy inequality. For the last inequality, we used (ii) in Theorem 7.3. As lim T →∞ h ( T ) =0 and ˆΓ u ( T ) is uniformly bounded in T, we obtain the desired result. C A note on condition (ii) in Theorem 7.1
This section discusses a method to analyze the derivative ∂∂η w η,ǫ ( x, T ) which is useful to check condition (ii) in Theorem 7.1. Appendices Dand E that discuss specific examples will rely on the following proposition. Proposition C.1.
Assume that φ η and f η are continuously differentiable in η on I. Fix
T > and assume the following conditions;(i) There exists a function g ( · , · ; T ) such that R T g ( X s , s ; T ) ds < ∞ a.s. and (cid:12)(cid:12)(cid:12) ∂∂η f η ( x, t ; T ) (cid:12)(cid:12)(cid:12) ≤ g ( x, t ; T ) for all η ∈ I, x ∈ ( ℓ, r ) and ≤ t ≤ T. (ii) There exists a random variable G T such that E Q [ G uT ] < ∞ for some u > and such that (cid:12)(cid:12)(cid:12) ∂φ η ∂η (cid:12)(cid:12)(cid:12) φ η ( X T ) e R T fη ( Xs,s ; T ) ds + 1 φ η ( X T ) e R T fη ( Xs,s ; T ) ds Z T (cid:12)(cid:12)(cid:12) ∂∂η f η ( X s , s ; T ) (cid:12)(cid:12)(cid:12) ds ≤ G T for all η ∈ I. Then ∂∂η w η,ǫ ( x, T ) = E Q ǫ (cid:20) ∂∂η (cid:16) φ η ( X ǫT ) e R T fη ( Xǫs,s ; T ) ds (cid:17)(cid:21) and ∂∂η w η,ǫ ( x, T ) is continuous in ( η, ǫ ) on I . Proof.
By direct calculation, it follows that ∂∂η (cid:16) φ η ( X T ) e R T fη ( Xs,s ; T ) ds (cid:17) = ∂φ η ∂η φ η ( X T ) e R T fη ( Xs,s ; T ) ds + 1 φ η ( X T ) e R T fη ( Xs,s ; T ) ds ∂∂η Z T f η ( X s , s ; T ) ds = ∂φ η ∂η φ η ( X T ) e R T fη ( Xs,s ; T ) ds + 1 φ η ( X T ) e R T fη ( Xs,s ; T ) ds Z T ∂∂η f η ( X s , s ; T ) ds. Condition (i) was used for the last equality in order to interchange the differentiation and integration using the Leibniz integral rule. Observethat w η,ǫ ( x, T ) = E Q ǫ (cid:20) φ η ( X ǫT ) e R T fη ( Xǫs,s ; T ) ds (cid:21) = E Q (cid:20) φ η ( X T ) e R T fη ( Xs,s ; T ) ds Z ǫT (cid:21) . From (ii), the Leibniz integral rule states that ∂∂η w η,ǫ ( x, T ) exists and ∂∂η w η,ǫ ( x, T ) = E Q (cid:20) ∂∂η (cid:16) φ η ( X T ) e R T fη ( Xs,s ; T ) ds (cid:17) Z ǫT (cid:21) . The continuity on I can be proven as follows. Using the same argument as in Step (II) of the proof of Proposition B.1, it suffices to showcontinuity at the origin ( η, ǫ ) = (0 , . Choose a sufficiently large even integer v and a sufficiently small u > /u + 1 /v = 1 . Define A ηT := ∂∂η (cid:16) φ η ( X T ) e R T fη ( Xs,s ; T ) ds (cid:17) ; A T := A T and we claim that E Q [ A ηT Z ǫT ] → E Q [ A T Z T ] as ( η, ǫ ) → (0 , . Using the inequalities (cid:12)(cid:12)(cid:12) E Q [ A ηT Z ǫT ] − E Q [ A T Z T ] (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) E Q [ A ηT ( Z ǫT − Z T )] (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) E Q [( A ηT − A T ) Z T ] (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) ( E Q (cid:12)(cid:12) A ηT (cid:12)(cid:12) u ) /u ( E Q | Z ǫT − Z T | v ) /v (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ( E Q (cid:12)(cid:12) A ηT − A T (cid:12)(cid:12) u ) /u ( E Q | Z T | v ) /v (cid:12)(cid:12)(cid:12) and since | A ηT | ≤ G T and E Q [ G uT ] < ∞ , it is enough to show that Z ǫT → Z T in L v as ǫ → . This was proven using Eq.(B.4) and the factthat lim ǫ → E Q [( Z ǫT ) i ] = 1 for 0 ≤ i ≤ v which was shown in Eq.(B.6). Finally, Girsanov’s theorem gives that ∂∂η w η,ǫ ( x, T ) = E Q (cid:20) ∂∂η (cid:16) φ η ( X T ) e R T fη ( Xs,s ; T ) ds (cid:17) Z ǫT (cid:21) = E Q ǫ (cid:20) ∂∂η (cid:16) φ η ( X ǫT ) e R T fη ( Xǫs,s ; T ) ds (cid:17)(cid:21) . The Kim–Omberg model
This appendix discusses the details of the Kim–Omberg model presented at Section 4.1, and shows the assumptions made in the main part ofthe paper are satisfied in this model. Assumptions A1 – 3 are well-known to be satisfied for the Kim–Omberg model. We recall the model inEq.(4.1) and investigate the corresponding objects v ( t, T ) , ˆ ξ, ( λ, φ ) , ξ ∗ , f, κ, Q . The function l ( ξ, x ) and h ( ξ, x ) in Eq.(5.5) are l ( ξ, x ) = − q − q ) (cid:16) µ x ς + ξ (cid:17) , h ( ξ, x ) = km − (cid:16) k + qµσ ς (cid:17) x − qσ ξ. The HJB equation (5.4) reads in this case v t = 12 σ v xx + sup ξ ∈ R { l ( ξ, x ) v + h ( ξ, x ) v x } = 12 σ v xx + (cid:18) km − (cid:16) k + qµσ ς (cid:17) x (cid:19) v x − q − q ) µ x ς v + qσ − q ) v x v with v ( x,
0) = 1 . Here, we used that the supremum of the above HJB equation is achieved at ξ = − σ − q v x v . The solution to this HJB equation corresponds to the function v in Eq.(5.2) (c.f. (Battauz et al., 2015, Lemma 3)) and can be expressed as v ( x, t ) = e Λ( t ) − β ( t ) x − γ ( t ) x , where the coefficients solve the following system of differential equations: β ′ ( t ) = − α β ( t ) − α β ( t ) + q (1 − q ) µ ς , β (0) = 0 ,γ ′ ( t ) = − ( α + α β ( t )) γ ( t ) + α β ( t ) , γ (0) = 0 , Λ ′ ( t ) = 12 α γ ( t ) − α γ ( t ) − σ β ( t, ) Λ(0) = 0 . (D.1)with α = k + qµσ ς , α = σ + σ − q , α = km, α = q α + q (1 − q ) α µ /ς . Thus assumption A4 holds. The first equation is the standard Riccati equation with solution β ( t ) = q (1 − q ) µ ς (1 − e − α t ) α + α + ( α − α ) e − α t . (D.2)Given β, the second equation of Eq.(D.1) is a first-order ODE which can be easily solved. The solution is γ ( t ) = α µ ( t ) Z t β ( s ) µ ( s ) ds where µ ( t ) = e R t α α β ( s )) ds . The optimal control in Eq.(5.7) ˆ ξ ( x, t ; T ) = − σ v x ( T − t, x )(1 − q ) v ( T − t, x ) = σ − q (cid:16) β ( T − t ) x + γ ( T − t ) (cid:17) (D.3)is obtained. With this optimizer, assumption A5 is satisfied by (Battauz et al., 2015, Eq.(26)).Now we shift our attention to the ergodic HJB equation (5.8). Direct calculation shows that φ ( x ) = e − Bx − Cx with the coefficients B = α − α α , C = α ( α − α ) α α , is a solution to the ergodic HJB equation (5.8). It is easy to show that β ( t ) → B, γ ( t ) → C and Λ( t ) t → − λ as t → ∞ , thus assumption A6holds. The optimal control ξ ∗ is given by ξ ∗ ( x ) = − σ φ x ( x )(1 − q ) φ ( x ) = σ − q (cid:0) Bx + C (cid:1) . (D.4)For the rest of this section, we show that assumptions A7 – A10 are satisfied. Proposition D.1.
For the Kim–Omberg model A7 holds, that is, the local martingale (cid:18) E (cid:16) − qµς Z · X s dW ,s − q Z · ˆ ξ ( X s , s ; T ) dW ,s (cid:17) t (cid:19) ≤ t ≤ T is a true martingale under the measure P . roof. In order to show this is a true martingale, we use Theorem 8.1 in Klebaner and Liptser (2014). Recall that dX t = k ( m − X t ) dt + σ dW ,t + σ dW ,t , X = χ. Using the notions in Klebaner and Liptser (2014), we have a t ( x ) = k ( m − x ) b t ( x ) = ( σ , σ ) ,σ t ( x ) = (cid:16) − qµxς , − q ˆ ξ ( x, t ; T ) (cid:17) , so that k σ t ( x ) k = q (cid:16) µ x ς + ˆ ξ ( x, t ; T ) (cid:17) ,L t ( x ) = 2 k ( m − x ) x + σ , L t ( x ) = − x (cid:16) − km + (cid:0) k + qµσ ς (cid:1) x + qσ ˆ ξ ( x, t ; T ) (cid:17) + σ . Using Eq.(D.3) and the fact that β ( T − t ) and γ ( T − t ) are bounded functions in t on [0 , T ] , one can find a positive r > χ = X such that k σ t ( x ) k + L t ( x ) + L t ( x ) ≤ r (1 + x ) . This implies that the assumptions of Theorem 8.1 in Klebaner and Liptser (2014) are met, and thus we obtain the desired result.Now the measure ˆ P is well-defined by Eq.(5.10) and the ˆ P -dynamics of X is dX t = ( km − ( k + qµσ ς ) X t − qσ ˆ ξ ( X t , t ; T )) dt + σ d ˆ W ,t + σ d ˆ W ,t . Proposition D.2.
For the Kim–Omberg model A8 holds, that is, the local martingale (cid:18) E (cid:16) q Z · ˆ ξ ( X s , s ; T ) − ξ ∗ ( X s ) d ˆ W ,s (cid:17) t (cid:19) ≤ t ≤ T is a true martingale under the measure ˆ P .Proof. In order to show this is a true martingale, we use Theorem 8.1 in Klebaner and Liptser (2014). The proof is similar to the proof ofProposition D.1, thus we only state the corresponding functions, a t ( x ) = km − ( k + qµσ ς ) x − qσ ˆ ξ ( x, t ; T ) ,b t ( x ) = ( σ , σ ) ,σ t ( x ) = (cid:0) , q (ˆ ξ ( x, t ; T ) − ξ ∗ ( x ) (cid:1) , and it is straightforward to verify that the assumptions of Theorem 8.1 in Klebaner and Liptser (2014) are met.Now the measure ˜ P is well-defined by Eq.(5.15) and the ˜ P -dynamics of X is dX t = (cid:16) km − ( k + qµσ ς ) X t − qσ ξ ∗ ( X t ) (cid:17) dt + σ d ˜ W ,t + σ d ˜ W ,t = (cid:18) km − qσ C − q − (cid:16) k + qµσ ς + qσ B − q (cid:17) X t (cid:19) dt + σ d ˜ W ,t + σ d ˜ W ,t , X = χ. Proposition D.3.
For the Kim–Omberg model, A9 holds, that is, the process M = (cid:18) E (cid:16) − Z · ( BX s + C ) σ d ˜ W ,s − Z · ( BX s + C ) σ d ˜ W ,s (cid:17) t (cid:19) ≤ t ≤ T is a martingale under the measure ˜ P .Proof. In order to show this is a true martingale, we use Theorem 8.1 in Klebaner and Liptser (2014). The proof is similar to the proof ofProposition D.1, thus we only state the corresponding functions, a t ( x ) = km − qσ C − q − (cid:16) k + qµσ ς + qσ B − q (cid:17) x,b t ( x ) = ( σ , σ ) ,σ t ( x ) = (cid:0) − σ ( Bx + C ) , − σ ( Bx + C ) (cid:1) , and it is straightforward to verify that the assumptions of Theorem 8.1 in Klebaner and Liptser (2014) are met.Now the measure P is well-defined by Eq.(5.16) and the P -dynamics of X is dX t = km − (cid:16) σ + σ − q (cid:17) C − (cid:18) k + qµσ ς + (cid:16) σ + σ − q (cid:17) B (cid:19) X t ! dt + σ dW ,t + σ dW ,t , (D.5)which is again the OU process with re-parametrization. 28 roposition D.4. For the Kim–Omberg model A10 holds, that is, the local martingale (cid:18) E (cid:16) q Z · ξ ∗ ( X s ) − ˆ ξ ( X s , s ; T ) dW ,s (cid:17) t (cid:19) ≤ t ≤ T is a true martingale under the measure P .Proof. In order to show this is a true martingale, we use Theorem 8.1 in Klebaner and Liptser (2014). The proof is similar to the proof ofProposition D.1, thus we only state the corresponding functions, a t ( x ) = km − (cid:16) σ + σ − q (cid:17) C − (cid:18) k + qµσ ς + (cid:16) σ + σ − q (cid:17) B (cid:19) x,b t ( x ) = ( σ , σ ) ,σ t ( x ) = (cid:16) , q (cid:0) ξ ∗ ( x ) − ˆ ξ ( x, t ; T ) (cid:1)(cid:17) , and it is straightforward to verify that the assumptions of Theorem 8.1 in Klebaner and Liptser (2014) are met.Now the measure Q is well-defined by Eq.(5.17) and the Q -dynamics of X is dX t = (cid:18) km − Cσ − qσ − q γ ( T − t ) − (cid:16) k + qµσ ς + Bσ + qσ − q β ( T − t ) (cid:17) X t (cid:19) dt + σ dB t (D.6)for 0 ≤ t ≤ T . The functions f and κ in Eq.(5.21) are f ( x, t ; T ) = − qσ − q ) (cid:16)(cid:0) B − β ( T − t ) (cid:1) x + (cid:0) C − γ ( T − t ) (cid:1)(cid:17) (D.7)and κ ( x, t ; T ) = km − Cσ − qσ − q γ ( T − t ) − (cid:16) k + qµσ ς + Bσ + qσ − q β ( T − t ) (cid:17) x. D.1 Integrability condition
In the following we prove integrability conditions, which will be needed in the analysis in the next sections.
Lemma D.5.
Let θ, σ be two positive constants and let W be a Brownian motion. Define Z t = σe − θt R t e θs dW s for t ≥ , which is thesolution of the SDE dZ t = − θZ t dt + σdW t , Z = 0 . For any α > and δ < αθσ , the expectation E [ e δe − αT R T eαsZ s ds ] is uniformly bounded for T ≥ . Proof. If δ ≤ , then the boundedness is trivial since the exponent is negative. Assume that 0 < δ < αθσ . Using the change of variable u = e αs , we get δe − αT Z T e αs Z s ds = 1 e αT − Z eαT δα (1 − e − αT ) Z u ) /α du. From Jensen’s inequality it follows that e δe − αT R T eαsZ s ds ≤ e αT − Z eαT e δα (1 − e − αT ) Z u ) /α du ≤ e αT − Z T αe αs e δα Z s ds. The random variable Z s is normally distributed with mean 0 and variance σ θ (1 − e − θt ) . Thus, for 0 < δ < αθσ , the expectation E [ e δα Z s ] isbounded on 0 ≤ s < ∞ . Let C be a positive number such that E [ e δα Z s ] ≤ C for all 0 ≤ s < ∞ . It follows that E [ e δe − αT R T eαsZ s ds ] ≤ e αT − Z T αe αs E [ e δα Z s ] ds ≤ Cαe αT − Z T e αs ds = C, which gives the desired result.We introduce the shorthand ζ t = ζ ( X t , t ; T ) = ξ ∗ ( X t ) − ˆ ξ ( X t , t ; T )to avoid a notationally heavy expression. From (D.5), the P -dynamics of X satisfies dX t = (cid:16) α α α − α X t (cid:17) dt + σ dW ,t + σ dW ,t which is a re-parametrized OU process. Lemma D.6.
For any δ < (1 − q ) α σ σ ( α + α ) ( α − α ) , the expectation E P [ e δ R T ζ Xs,s ; T ) ds ] is uniformly bounded in T ≥ . roof. Define a := α α /α , and a process W := σ σ W + σ σ W so that the process X satisfies dX t = α ( a − X t ) dt + σ dW t , X = χ. The solution of this SDE is X t = χe − α t + a (1 − e − α t ) + Z t where Z t = σe − α t R t e α s dW s . From Eq.(D.3) and (D.4), it can be shown that ζ ( x, t ; T ) = ξ ∗ ( x ) − ˆ ξ ( x, t ; T ) = σ − q (cid:16) ( B − β ( T − t )) x + C − γ ( T − t ) (cid:17) and it is easy to show that | B − β ( t ) | ≤ α ( α − α ) α ( α + α ) e − α t , | C − γ ( t ) | ≤ c e − α t (D.8)for some positive constant c . For the second inequality, we observe thatlim t →∞ γ ( t ) − Ce − α t = lim t →∞ α R t β ( s ) µ ( s ) ds − Cµ ( t ) µ ( t ) e − α t = lim t →∞ ( α − Cα )( β ( t ) − B )( α + α β ( t ) − α ) e − α t and the limit converges to a nonzero constant. Here, we used α B − C ( α + α B ) = 0 , L’Hˆopital’s rule and Eq.(D.2). Then ζ ( x, t ; T ) ≤ c e − α T − t ) x + (const) e − α T − t ) x + (const) e − α T − t ) where c := 2 σ α ( α − α )(1 − q ) α ( α + α ) . The large-time behavior of the expectation E P [ e δ R T ζ Xs,s ; T ) ds ] depends only on the highest-order term c e − α T − t ) X t . Using that X t ≤ Z t + x + a , it suffices to prove that for such a δ the expectation E P e δc e − α T R T e α sZ s ds is uniformly bounded in T ≥ . Lemma D.5 gives that this expectation is uniformly bounded in T ≥ δc < α σ , which gives the desiredresult. Lemma D.7.
There are positive numbers c and r > such that for any T ≥ and any nonnegative path functiona h E Q [ h ( X ·∧ T )] ≤ c (cid:0) E P [ h r ( X ·∧ T )] (cid:1) /r . We emphasize that the positive constants c and r do not depend on the time T ≥ h. Proof.
One can first find a positive δ such that E P e δq R T ζ s ds is uniformly bounded in T ≥ r > r > δ = r ( r − , and define r > r + r + r = 1 . Then E Q [ h ( X ·∧ T )] = E P h h ( X ·∧ T ) e q R T ζs dW ,s − q R T ζ s ds i ≤ (cid:0) E P [ h r ( X ·∧ T )] (cid:1) r (cid:18) E P h e r r − q R T ζ s ds i(cid:19) r (cid:18) E P h e r q R T ζs dW ,s − r q R T ζ s ds i(cid:19) r . The last term is a positive local martingale so that the expectation is less than or equal to 1 . It follows that E Q [ h ( X ·∧ T )] ≤ (cid:0) E P [ h r ( X ·∧ T )] (cid:1) r (cid:0) E P [ e δq R T ζ s ds ] (cid:1) r The second term E P [ e δq R T ζ s ds ] is uniformly bounded in T ≥ δ. This gives the desired result.
Lemma D.8.
For any δ > , the expectation E Q (cid:2) | X T | δ (cid:3) is uniformly bounded in ( x, T ) on ( χ − , χ + 1) × [0 , ∞ ) . Proof.
From Lemma D.7, there are positive numbers c and r > T ≥ E Q (cid:2) | X T | δ (cid:3) ≤ c (cid:16) E P (cid:2) | X T | rδ (cid:3)(cid:17) /r . The right-hand side is uniformly bounded in ( x, T ) on ( χ − , χ + 1) × [0 , ∞ ) since X is an OU process under the measure P . Lemma D.9.
There are a number u > and an open neighborhood I χ of χ such that Γ u ( x, T ) := E Q h φ u ( X T ) e u R T f ( Xs,s ; T ) ds i is uniformly bounded on I χ × [0 , ∞ ) . roof. Since the function f is nonpositive as one can see in Eq.(D.7), it suffices to show that there is a number u > E Q h φ u ( X T ) i = E Q h φ u ( X T ) (cid:12)(cid:12)(cid:12) X = x i is uniformly bounded in ( x, T ) on ( χ − , χ + 1) × [0 , ∞ ) . Define β Q ( t ) := k + qµσ ς + Bσ + qσ − q β ( t ) , γ Q ( t ) := km − Cσ − qσ − q γ ( t ) , then the Q -dynamics of X is dX t = (cid:0) γ Q ( T − t ) − β Q ( T − t ) X t (cid:1) dt + σ dB t , X = x for 0 ≤ t ≤ T . Solving this SDE, it follows that X T = xe − R T β Q ( T − s ) ds + e − R T β Q ( T − u ) du Z T γ Q ( T − s ) e R s β Q ( T − u ) du ds + σe − R T β Q ( T − u ) du Z T e R s β Q ( T − u ) du dB s . The random variable X T is normally distributed with mean m T = xe − R T β Q ( T − s ) ds + e − R T β Q ( T − s ) ds Z T γ Q ( T − s ) e R s β Q ( T − u ) du ds = xe − R T β Q ( s ) ds + Z T γ Q ( s ) e − R s β Q ( u ) du ds and variance v T = σ e − R T β Q ( T − u ) du Z T e R s β Q ( T − u ) du ds = σ Z T e − R s β Q ( u ) du ds. In addition, it is easy to check the limits exist, i.e., m ∞ := lim T →∞ m T = Z ∞ γ Q ( s ) e − R s β Q ( u ) du ds, v ∞ := lim T →∞ v T = σ Z ∞ e − R s β Q ( u ) du ds. The Q -density function of X T is 1(2 πv T ) / e −
12 ( x − mT )2 v T , thus E Q h φ u ( X T ) i = E Q h e uBX T + uCXT i = 1(2 πv T ) / Z ∞−∞ e uBz uCz −
12 ( z − mT )2 v T dz. (D.9)Observe that β Q ( t ) ≥ k + qµσ ς + Bσ . We have v T ≤ v ∞ = σ Z ∞ e − R s β Q ( u ) du ds ≤ σ Z ∞ e − k + qµσ ς + Bσ s ds = σ k + qµσ ς + Bσ ) . The integral in Eq.(D.9) satisfies Z ∞−∞ e uBz uCz −
12 ( z − mT )2 v T dz ≤ Z ∞−∞ e uBz uCz − σ k + qµσ ς + Bσ z − mT )2 dz. (D.10)Using the condition k + qµσ ς + Bσ > , one can choose a small u > x, T ) on( χ − , χ + 1) × [0 , ∞ ) . D.2 Sensitivity with respect to the initial volatility
The purpose of this section is to prove the following proposition, which yields the first statement of Theorem 4.1.
Proposition D.10.
For the Kim–Omberg model presented in Eq. (4.1) , the long-term sensitivity with respect to the initial value of thevolatility is lim T →∞ ∂∂χ ln v ( χ, T ) = − Bχ − C. Proof.
By Theorem 3.2, it suffices to prove that the expectation E Q (cid:2) φ ( XT ) e R T f ( Xs,s ; T ) ds (cid:12)(cid:12) X = x (cid:3) is continuously differentiable in x, and ∂∂x E Q h φ ( X T ) e R T f ( Xs,s ; T ) ds (cid:12)(cid:12)(cid:12) X = x i converges to zero as T → ∞ . To prove this, we apply Proposition 6.1. Condition (i) of this proposition was proved in Lemma D.9. For (ii),we fix any v > . By Lemma D.8, it follows that E Q (cid:12)(cid:12)(cid:12)(cid:12) φ ′ ( X T ) φ ( X T ) (cid:12)(cid:12)(cid:12)(cid:12) v = E Q | BX T + C | v is uniformly bounded in ( x, T ) on ( χ − , χ + 1) × [0 , ∞ ) . To show (iii), we calculate the first variation process Y of X given Eq.(D.6). Then Y t = Y t ; T satisfies dY t = − (cid:16) k + qµσ ς + Bσ + qσ − q β ( T − t ) (cid:17) Y t dt, Y = 1 , ≤ t ≤ T, which is a deterministic process. It follows that Y t ; T = e − ( k + qµσ ς + Bσ t − qσ − q R t β ( T − s ) ds .
31y direct calculation, for any fixed w > , it is clear thatlim T →∞ E Q | Y T ; T | w = lim T →∞ e − w ( k + qµσ µ + Bσ T − w qσ − q R T β ( T − s ) ds = 0since k + qµσ µ + Bσ > β ( · ) > . We now consider (iv). By using Eq.(D.8), it can be easily shown that there are positive constants c and c such that (cid:12)(cid:12) f x ( x, t ; T ) (cid:12)(cid:12) = qσ − q (cid:12)(cid:12)(cid:12)(cid:12)(cid:16)(cid:0) B − β ( T − t ) (cid:1) x + (cid:0) C − γ ( T − t ) (cid:1)(cid:17)(cid:16) B − β ( T − t ) (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) ≤ c e − c T − t ) ( | x | + 1) . By using Y t ; T ≤ e − νt where ν := k + qµσ ς + Bσ , we obtain that for any m > E Q (cid:20)(cid:18)Z T | f x ( X s , s ; T ) Y s ; T | ds (cid:19) m (cid:21) ≤ c m T m − e − c mT Z T e ( c − ν ) ms E Q (cid:2) ( | X s | + 1) m (cid:3) ds by Jensen’s inequality. Using Lemma D.8, we observe that for each m > , the expectation E Q [( | X s | + 1) m ] is uniformly bounded in s ≥ C m . Thus, E Q (cid:20)(cid:18)Z T | f x ( X s , s ; T ) Y s ; T | ds (cid:19) m (cid:21) ≤ c m C m ( c − ν ) m T m − (cid:16) e − νmT − e − c mT (cid:17) → T → ∞ . Finally, conditions (ii), (iii), (iv) in Proposition 6.1 hold true for arbitrary v, w, m > , and (i) holds for some u > , so we obtainthe desired result. D.3 Sensitivities with respect to k, m, µ, ς and ρ We compute the long-term sensitivity with respect to the perturbation of k. Those with respect to the parameters m, µ, ς and ρ can becalculated in a similar way because all these parameters affect the functionals φ, f and the drift of X but not the volatility of X as seen inthe Q -dynamics of XdX t = (cid:18) km − Cσ − q (1 − ρ ) σ − q γ ( T − t ) − (cid:16) k + qµρσς + Bσ + q (1 − ρ ) σ − q β ( T − t ) (cid:17) X t (cid:19) dt + σ dB t for 0 ≤ t ≤ T . The five functions in B1 and B2 are m ǫ ( x ) = ( k + ǫ )( m − x ) , σ ,ǫ ( x ) = σ , σ ,ǫ ( x ) = σ , b ǫ ( x ) = µx, ς ǫ ( x ) = ς and it is easy to check that they satisfy assumptions B1 and B2. Observe that ∂∂ǫ (cid:12)(cid:12)(cid:12) ǫ =0 ln v ǫ ( χ, T ) = ∂∂k ln v ( χ, T ) = ∂∂k ln v ( χ, T ) , thus for the rest of this section we use ∂∂k instead of ∂∂ǫ | ǫ =0 . Lemma D.11.
Let α > and ℓ > . The expectation E Q (cid:20)(cid:18)Z T e − α ( T − s ) X s ds (cid:19) ℓ (cid:21) is uniformly bounded in T on [0 , ∞ ) . Proof.
By Lemma D.7, there are positive numbers c and r, independent of T, such that E Q (cid:20)(cid:18) Z T e − α ( T − s ) X s ds (cid:19) ℓ (cid:21) ≤ c E P (cid:20)(cid:18) Z T e − α ( T − s ) X s ds (cid:19) rℓ (cid:21)! /r . From Lemma D.5, we know that E P h e δ R T e − α ( T − s ) X s ds i is uniformly bounded in T for sufficiently small δ > . Choose n ∈ N such that rℓ ≤ n. Using the inequality xnn ! ≤ e x for x > , we have δ n n ! E P (cid:20)(cid:18)Z T e − α ( T − s ) X s ds (cid:19) rℓ (cid:21) ≤ δ n n ! E P (cid:20)(cid:18)Z T e − α ( T − s ) X s ds (cid:19) n (cid:21) ≤ E P h e δ R T e − α ( T − s ) X s ds i . Thus, E P (cid:20)(cid:18)Z T e − α ( T − s ) X s ds (cid:19) rℓ (cid:21) is also uniformly bounded in T on [0 , ∞ ) , which gives the desired result. Proposition D.12.
For the Kim–Omberg model presented in Eq. (4.1) , the long-term sensitivity with respect to the parameter k is lim T →∞ T ∂∂k ln v ( χ, T ) = − ∂λ∂k . roof. To prove this equality, we use Theorem 7.1. Condition (i) in Theorem 7.1 is satisfied trivially. We prove (iii) in Theorem 7.1 firstbecause some techniques used for (iii) are also used in the proof of (ii). For condition (iii) in Theorem 7.1, we apply Theorem 7.3. It can beeasily checked that (cid:12)(cid:12)(cid:12) ∂∂k κ ( x, t ; T ) (cid:12)(cid:12)(cid:12) ≤ c ( | x | + 1)for a positive constant c independent of t, T and x. By choosing sufficiently large c, we can achieve that ˆ g ( x, t ; T ) ≤ c ( | x | + 1) holds true forˆ g defined in Eq.(7.4).Then, (i) in Theorem 7.3 can be proven as follows. Since X is an OU process under the measure P , for each T > δ = δ ( T ) such that E P h e δ R T X s ds i is finite. For the positive constant r in Lemma D.7, we define ǫ = δ c r , then E Q h e ǫ R T g Xs,s ; T ) ds i ≤ E Q h e ǫ c R T | Xs | +1)2 ds i ≤ c ′ (cid:18) E P h e ǫ c r R T | Xs | +1)2 ds i(cid:19) /r ≤ c ′ (cid:18) E P h e ǫ c r R T X s +1) ds i(cid:19) /r = c ′ e ǫ c T (cid:18) E P h e ǫ c r R T X s ds i(cid:19) /r = c ′ e ǫ c T (cid:18) E P h e δ R T X s ds i(cid:19) /r where c ′ is the positive constant in Lemma D.7. This gives (i) in Theorem 7.3.For (ii) in Theorem 7.3, we observe that for any v ≥ E Q (cid:20)(cid:16)Z T ˆ g ( X s , s ; T ) ds (cid:17) v/ (cid:21) ≤ c v E Q (cid:20)(cid:16)Z T ( | X s | + 1) ds (cid:17) v/ (cid:21) ≤ c v T v/ E Q (cid:20)(cid:16) T Z T ( | X s | + 1) ds (cid:17) v/ (cid:21)! ≤ c v T v/ (cid:18) E Q h T Z T ( | X s | + 1) v ds i(cid:19) = c v T v/ − (cid:18)Z T E Q [( | X s | + 1) v ] ds (cid:19) . By Lemma D.8, the expectation E Q [( | X s | + 1) v ] is uniformly bounded in s by a positive constant, say C. Then E Q (cid:20)(cid:16)Z T ˆ g ( X s , s ; T ) ds (cid:17) v/ (cid:21) ≤ c v T v/ − (cid:18)Z T E Q [( | X s | + 1) v ] ds (cid:19) ≤ c v CT v/ . Since the constants c and C do not depend on T, we obtain the desired result. For (iii) in Theorem 7.3, we observe that for ǫ = 1 E Q (cid:20)Z T ˆ g v + ǫ ( X s , s ; T ) ds (cid:21) ≤ c v +1 Z T E Q h ( | X s | + 1) v +1 i ds, and the right-hand side is finite for each T ≥ E Q (cid:2) ( | X s | + 1) v +1 (cid:3) is uniformly bounded in s by Lemma D.8.For (iv) Theorem 7.3, we want to show that for u with 1 /u + 1 /v = 1 the expectation E Q h φ u ( X T ) e u R T f ( Xs,s ; T ) ds i is uniformly bounded in T on [0 , ∞ ) . However, observe that we proved that (ii) and (iii) in Theorem 7.3 hold true for arbitrary v ≥ . Thus,it is enough to show that such u > B ( k ) and C ( k ) to emphasize the dependence of k on the constants B and C, respectively. From Eq.(D.9) and Eq.(D.10), we know for a small u > E Q h e u B ( k ) X T + u C ( k ) XT i (D.11)is uniformly bounded in T on [0 , ∞ ) . Since the two maps k B ( k ) and k C ( k ) are continuous and u > , by choosing a smallerinterval I if necessary, it follows that sup ǫ ∈ I B ( k + ǫ ) ≤ u + 12 B ( k ) , sup ǫ ∈ I C ( k + ǫ ) ≤ u + 12 C ( k ) . Then ˆ φ ( x ) = inf ǫ ∈ I e − B ( k + ǫ ) x − C ( k + ǫ ) x ≥ e − u B ( k ) x − u C ( k ) x . (D.12)Define ˆ u := 2 u u + 1 > , (D.13)then we have E Q h φ ˆ u ( X T ) e ˆ u R T f ( Xs,s ; T ) ds i ≤ E Q h φ ˆ u ( X T ) i ≤ E Q h e u B ( k ) X T + u C ( k ) XT i (D.14)where for the first inequality we used ˆ f ≤ . Since the right-hand side is uniformly bounded in T on [0 , ∞ ) , we obtain the desired result. Wehave now shown all conditions in Theorem 7.3 and thus condition (iii) in Theorem 7.1 holds true.For condition (ii) in Theorem 7.1, we first calculate the partial derivative with respect to the variable k in φ and f but not in X = ( X t ) t ≥ . To be precise, we use notation φ ( x ; k ) and f ( x, t ; T ; k ) to emphasize the dependence of k. We want to analyze w η,ǫ ( χ, T ) = E Q ǫ h φ ( X ǫT ; k + η ) e R T f ( Xǫs,s ; T ; k + η ) ds i where the Q ǫ -dynamics of X ǫt satisfies Eq.(D.6) with k replaced by k + ǫ. The equality ∂∂η E Q h φ ( X ǫT ; k + η ) e R T f ( Xǫs,s ; T ; k + η ) ds i = E Q (cid:20) ∂∂η (cid:16) φ ( X ǫT ; k + η ) e R T f ( Xǫs,s ; T ; k + η ) ds (cid:17)(cid:21) η, ǫ ) on I are obtained from Proposition C.1 with g ( x, t ; T ) and G T given below. Observethat ∂f∂k ( x, t ; T ; k ) = − qσ − q (cid:16)(cid:0) B − β ( T − t ) (cid:1) x + (cid:0) C − γ ( T − t ) (cid:1)(cid:17)(cid:18)(cid:16) ∂B∂k − ∂β∂k ( T − t ) (cid:17) x + (cid:16) ∂C∂k − ∂γ∂k ( T − t ) (cid:17)(cid:19) . We use the notations β ( T − t ; k ) , γ ( T − t ; k ) to emphasize the dependence of k. For a given small open interval I, since B ( k + η ) , C ( k + η ) , ∂B∂k ( k + η ) , ∂C∂k ( k + η ) are continuous in η on I and β ( T − t ; k + η ) , γ ( T − t ; k + η ) , ∂β∂k ( T − t ; k + η ) , ∂γ∂k ( T − t ; k + η ) are continuous in( η, t ) on I × [0 , T ] , one can find a positive constant b such that for all ( η, t ) ∈ I × [0 , T ] (cid:12)(cid:12)(cid:12)(cid:12) ∂f∂η ( x, t ; T ; k + η ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ b ( x + 1) =: g ( t, x ; T ) . With this function g, condition (i) in Proposition C.1 is trivially satisfied. For condition (ii) in Proposition C.1, choose two positive constants b and c such that for all η ∈ I (cid:12)(cid:12)(cid:12)(cid:12) ∂B∂η ( k + η ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ b , (cid:12)(cid:12)(cid:12)(cid:12) ∂C∂η ( k + η ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ c . Using the function ˆ φ in Eq.(D.12), we define G T := 1ˆ φ ( X T ) (cid:16) b X T + c | X T | (cid:17) + 1ˆ φ ( X T ) Z T b ( X s + 1) ds. Then for all ( η, t ) ∈ I × [0 , T ] it follows that1 φ ( X T ; k + η ) (cid:12)(cid:12)(cid:12)(cid:12) ∂φ∂η ( X T ; k + η ) (cid:12)(cid:12)(cid:12)(cid:12) + 1 φ ( X T ; k + η ) Z T (cid:12)(cid:12)(cid:12)(cid:12) ∂f∂η ( X s , s ; T ; k + η ) (cid:12)(cid:12)(cid:12)(cid:12) ds ≤ G T by using that ˆ φ ( x ) = inf η ∈ I φ ( x ; k + η ) and (cid:12)(cid:12)(cid:12) ∂φ∂η ( x ; k + η ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) x ∂B∂η ( k + η ) + x ∂C∂η ( k + η ) (cid:12)(cid:12)(cid:12) φ ( x ; k + η ) ≤ (cid:16) b x + c | x | (cid:17) φ ( x ; k + η ) . Recall u > φ ( x ) ≥ e − u B ( k ) x − u C ( k ) x and ˆ u = u u > E Q [ G u T ] < ∞ for u = u u > , which implies condition (ii) in Proposition C.1. Let ˆ v be such that u/u + v = 1, then E Q h φ u ( X T ) (cid:16) b X T + c | X T | (cid:17) u i ≤ (cid:18) E Q h φ ˆ u ( X T ) i(cid:19) u / ˆ u (cid:18) E Q h(cid:16) b X T + c | X T | (cid:17) u v i(cid:19) / ˆ v . The two expectations on the right-hand side are finite by Eq.(D.14) and Lemma D.8. In a similar way, we have E Q (cid:20) φ u ( X T ) (cid:16) b Z T ( X s + 1) ds (cid:17) u (cid:21) ≤ (cid:18) E Q h φ ˆ u ( X T ) i(cid:19) u / ˆ u (cid:18) E Q h(cid:16) b Z T ( X s + 1) ds (cid:17) u v i(cid:19) / ˆ v ≤ T u − v (cid:18) E Q h φ ˆ u ( X T ) i(cid:19) u / ˆ u (cid:18) E Q h b u v Z T ( X s + 1) u v ds i(cid:19) / ˆ v ≤ T u − v (cid:18) E Q h φ ˆ u ( X T ) i(cid:19) u / ˆ u (cid:18) b u v Z T E Q (cid:2) ( X s + 1) u v (cid:3) ds (cid:19) / ˆ v . Since E Q [( X s + 1) u v ] is uniformly bounded in s on [0 , ∞ ) byD.8, the right-hand side is finite. Hence, E Q [ G u T ] < ∞ . The convergence lim T →∞ T ∂∂η (cid:12)(cid:12)(cid:12) η =0 E Q h φ ( X T ; k + η ) e R T f ( Xs,s ; T ; k + η ) ds i = 0can be shown as follows. The partial derivative with respect to η satisfies (cid:12)(cid:12)(cid:12)(cid:12) ∂∂η (cid:12)(cid:12)(cid:12) η =0 (cid:16) φ ( X T ; k + η ) e R T f ( Xs,s ; T ; k + η ) ds (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) ≤ e BX T + CXT + R T f ( Xs,s ; T ; k ) ds (cid:12)(cid:12)(cid:12) X T ∂B∂k + X T ∂C∂k (cid:12)(cid:12)(cid:12) + e BX T + CXT + R T f ( Xs,s ; T ; k ) ds (cid:12)(cid:12)(cid:12)(cid:12)Z T ∂f∂η (cid:12)(cid:12)(cid:12) η =0 ( X s , s ; T ; k + η ) ds (cid:12)(cid:12)(cid:12)(cid:12) ≤ e BX T + CXT (cid:12)(cid:12)(cid:12) X T ∂B∂k + X T ∂C∂k (cid:12)(cid:12)(cid:12) + e BX T + CXT (cid:12)(cid:12)(cid:12)(cid:12)Z T ∂f∂η (cid:12)(cid:12)(cid:12) η =0 ( X s , s ; T ; k + η ) ds (cid:12)(cid:12)(cid:12)(cid:12) . By the triangle inequality and the H¨older inequality, for u in Eq.(D.11) and v satisfying 1 /u + 1 /v = 1 it follows that E Q (cid:12)(cid:12)(cid:12)(cid:12) ∂∂η (cid:12)(cid:12)(cid:12) η =0 (cid:16) φ ( X T ; k + η ) e R T f ( Xs,s ; T ; k + η ) ds (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:16) E Q e u BX T + u CXT (cid:17) /u (cid:16) E Q (cid:12)(cid:12)(cid:12) X T ∂B∂k + X T ∂C∂k (cid:12)(cid:12)(cid:12) v (cid:17) /v + (cid:16) E Q e u BX T + u CXT (cid:17) /u (cid:18) E Q (cid:12)(cid:12)(cid:12)Z T ∂f∂η (cid:12)(cid:12)(cid:12) η =0 ( X s , s ; T ; k + η ) ds (cid:12)(cid:12)(cid:12) v (cid:19) /v . By the choice of u , the expectation E Q e u BX T + u CXT is uniformly bounded in T. The expectation E Q | X T ∂B∂k + X T ∂C∂k | v is alsouniformly bounded in T by Lemma D.8. Now, we show that the expectation E Q | R T ∂f∂η | η =0 ( X s , s ; T ; k + η ) ds | v is uniformly bounded in T .By direct calculation, one can choose positive constants c and d, which are independent of s and T but are dependent of k, such that (cid:12)(cid:12)(cid:12)(cid:12) ∂f∂η (cid:12)(cid:12)(cid:12) η =0 ( x, s ; T ; k + η ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ de − c ( T − s ) ( x + 1) . u = e cs , observe that E Q h(cid:16)Z T e cs ( X s + 1) ds (cid:17) v i = E Q h(cid:16)Z ecT c ( X u ) /c + 1) du (cid:17) v i = ( e cT − v c v E Q h(cid:16) e cT − Z ecT ( X u ) /c + 1) du (cid:17) v i ≤ ( e cT − v c v E Q h e cT − Z ecT ( X u ) /c + 1) v du i = ( e cT − v − c v Z ecT E Q h(cid:0) X u ) /c + 1 (cid:1) v i du. (D.15)By Lemma D.8, there is a positive constant C such that E Q [( X u ) /c + 1) v ] ≤ C for all u ≥ . Thus, E Q (cid:12)(cid:12)(cid:12)(cid:12)Z T ∂f∂η (cid:12)(cid:12)(cid:12) η =0 ( X s , s ; T ; k + η ) ds (cid:12)(cid:12)(cid:12)(cid:12) v ≤ d v e − cv T E Q h(cid:16)Z T e cs ( X s + 1) ds (cid:17) v i ≤ d v e − cv T ( e cT − v − c v Z ecT E Q h(cid:0) X u ) /c + 1 (cid:1) v i du ≤ d v e − cv T ( e cT − v − c v Z ecT C du ≤ Cd v c v , (D.16)which gives the desired result. D.4 Sensitivity with respect to σ We evaluate the long-term sensitivity with respect to the perturbations of σ. Proposition D.13.
Under the Kim–Omberg model in Eq. (4.1) , the long-term sensitivity with respect to the parameter σ is lim T →∞ T ∂∂σ ln v ( χ, T ) = − ∂λ∂σ . Proof.
In the decomposition v ( χ, T ) = e − λT φ ( χ ) E Q h φ ( X T ) e R T f ( Xs,s ; T ) ds i , we analyze the expectation term E Q [ φ ( XT ) e R T f ( Xs,s ; T ) ds ] by using the method in Section 7.3. Consider the Lamperti transformation ℓ ( x ) = Z xχ σ du = x − χσ . and define ˇ X t := ℓ ( X t ) = X t − χσ as well as F (ˇ x ) = − qσ − q ) (cid:16)(cid:0) B − β ( T − t ) (cid:1) ( σ ˇ x + χ ) + (cid:0) C − γ ( T − t ) (cid:1)(cid:17) , Φ(ˇ x ) = e − Bσ x − ( Bχ + C ) σ ˇ x − Bχ − Cχ . Then ˇ X satisfies the SDE d ˇ X t = (cid:18) σ (cid:0) km − Cσ − qσ − q γ ( T − t ) (cid:1) − (cid:16) k + qµσ ς + Bσ + qσ − q β ( T − t ) (cid:17)(cid:0) ˇ X t + χσ (cid:1)(cid:19) dt + dB t . We want to analyze ∂∂σ E Q h X T ) e R T F ( ˇ Xs,s ; T ) ds i . The perturbation parameter σ is only involved with the functional and the drift term of ˇ X , but not with the volatility term of ˇ X. Thus, wecan apply the same method used in Proposition D.12 to showlim T →∞ T ∂∂σ E Q h X T ) e R T F ( ˇ Xs,s ; T ) ds i = 0 . This gives the desired result.
E The Heston model
This appendix investigates the Heston model presented in Section 4.2 and shows the assumptions made in the main part of the paper aresatisfied in this model. Assumptions A1 – 3 are well-known to be satisfied for the Heston model.We first find the HJB equation and the ergodic HJB equation. The functions l and h in Eq.(5.5) are l ( ξ, x ) := − q − q ) (cid:16) µ xς + ξ (cid:17) h ( ξ, x ) := km − (cid:16) k + qµσ ς (cid:17) x − qξσ √ x. v t = 12 σ xv xx + sup ξ ∈ R ( − q − q ) (cid:16) µ xς + ξ (cid:17) v + (cid:18) km − (cid:16) k + qµσ ς (cid:17) x − qξσ √ x (cid:19) v x ) = 12 σ xv xx − q − q ) µ ς xv + (cid:18) km − (cid:16) k + qµσ ς (cid:17) x (cid:19) v x + qσ x − q ) v x v with v ( x,
0) = 1. Here, we used that the supremum of the above HJB equation is achieved at ξ = − σ √ x (1 − q ) v x ( x, t ) v ( x, t ) . The solution to the HJB equation is v ( x, t ) = e − γ ( t ) − β ( t ) x with β ( t ) = q (1 − q ) µ ς sinh( β t/ β cosh( β t/
2) + β sinh( β t/ ,γ ( t ) = km Z t B ( s ) ds, (E.1)where β := k + qµσ ς , β := s β + q ((1 − q ) σ + σ ) µ ς . Thus assumption A4 holds. The optimal control ˆ ξ is ˆ ξ ( x, t ; T ) = σ − q β ( T − t ) √ x (E.2)With this optimizer, assumption A5 is satisfied.Now we shift our attention to the ergodic HJB equation (5.8). By direct calculation, we can see that the solution to the ergodic HJBequation − λφ = 12 σ xφ xx − q − q ) µ xφ + (cid:0) km − ( qµσ + k ) x (cid:1) φ x + qσ x − q ) φ x φ is given by φ ( x ) = e − Bx with B = β − β σ + σ − q . It is easy to show that β ( t ) → B and γ ( t ) t → − λ as t → ∞ , thus assumption A6 holds. The ergodic optimal control ξ ∗ is ξ ∗ ( x ) = σ − q B √ x. For the rest of this section, we show that assumptions A7 – A10 are satisfied.
Proposition E.1.
For the Heston model A7 holds, that is, the local martingale (cid:18) E (cid:16) − qµς Z · p X s dW ,s − q Z · ˆ ξ ( X s , s ; T ) dW ,s (cid:17) t (cid:19) ≤ t ≤ T is a true martingale under the measure P .Proof. In order to show this is a true martingale, we use Theorem 8.1 in Klebaner and Liptser (2014). Recall that dX t = k ( m − X t ) dt + σ p X t dW ,t + σ p X t dW ,t , X = χ. Using the notions in Klebaner and Liptser (2014), we have a t ( x ) = k ( m − x ) ,b t ( x ) = (cid:0) σ √ x, σ √ x (cid:1) ,σ t ( x ) = (cid:16) − qµς √ x, − q ˆ ξ ( x, t ; T ) (cid:17) , so that k σ t ( x ) k = q (cid:16) µ xς + ˆ ξ ( x, t ; T ) (cid:17) ,L t ( x ) = 2 k ( m − x ) x + σ x, L t ( x ) = 2 x (cid:18) km − (cid:16) k + qµσ ς (cid:17) x − qσ ˆ ξ ( x, t ; T ) √ x (cid:19) + σ x. Using the expression of ˆ ξ in Eq.(E.2), one can find a positive r > χ = X such that k σ t ( x ) k + L t ( x ) + L t ( x ) ≤ r (cid:0) x (cid:1) . This implies that the assumptions of Theorem 8.1 in Klebaner and Liptser (2014) are met, and thus we obtain the desired result.36ow the measure ˆ P is well-defined by Eq.(5.10) and the ˆ P -dynamics of X is dX t = (cid:18) km − (cid:16) k + qµσ ς (cid:17) X t − qσ ˆ ξ ( X t , t ; T ) p X t (cid:19) dt + σ p X t d ˆ W ,t + σ p X t d ˆ W ,t , X = χ. Proposition E.2.
For the Heston model A8 holds, that is, the local martingale (cid:18) E (cid:16) q Z · ˆ ξ ( X s , s ; T ) − ξ ∗ ( X s ) d ˆ W ,s (cid:17) t (cid:19) ≤ t ≤ T is a true martingale under the measure ˆ P .Proof. In order to show this is a true martingale, we use Theorem 8.1 in Klebaner and Liptser (2014). The proof is similar to the proof ofProposition E.1, thus we only state the corresponding functions, a t ( x ) = km − (cid:16) k + qµσ ς (cid:17) x − qσ ˆ ξ ( x, t ; T ) √ x,b t ( x ) = (cid:0) σ √ x, σ √ x (cid:1) ,σ t ( x ) = (cid:16) , q (cid:0) ˆ ξ ( x, t ; T ) − ξ ∗ ( x ) (cid:1)(cid:17) , and it is straightforward to verify that the assumptions of Theorem 8.1 in Klebaner and Liptser (2014) are met.Now the measure ˜ P is well-defined by Eq.(5.15) and the ˜ P -dynamics of X is dX t = (cid:18) km − (cid:16) k + qµσ ς (cid:17) X t − qσ ξ ∗ ( X t ) p X t (cid:19) dt + σ p X t d ˜ W ,t + σ p X t d ˜ W ,t , X = χ. Proposition E.3.
For the Heston model, A9 holds, that is, the process M = (cid:18) E (cid:16) − Z · σ B p X s d ˜ W ,s − Z · σ B p X s d ˜ W ,s (cid:17) t (cid:19) ≤ t ≤ T is a martingale under the measure ˜ P .Proof. In order to show this is a true martingale, we use Theorem 8.1 in Klebaner and Liptser (2014).The proof is similar to the proof ofProposition E.1, thus we only state the corresponding functions, a t ( x ) = km − (cid:16) k + qµσ ς (cid:17) x − qσ ξ ∗ ( x ) √ x,b t ( x ) = (cid:0) σ √ x, σ √ x (cid:1) σ t ( x ) = (cid:0) − σ B √ x, − σ B √ x (cid:1) , and it is straightforward to verify that the assumptions of Theorem 8.1 in Klebaner and Liptser (2014) are met.Now the measure P is well-defined by Eq.(5.16) and the P -dynamics of X is dX t = (cid:18) km − (cid:16) k + qµσ ς + σ B (cid:17) X t − qσ ξ ∗ ( X t ) p X t (cid:19) dt + σ p X t dW ,t + σ p X t dW ,t = km − (cid:18) k + qµσ ς + (cid:16) σ + σ − q (cid:17) B (cid:19) X t ! dt + σ p X t dW ,t + σ p X t dW ,t which is again the CIR process with re-parametrization. Proposition E.4.
For the Heston model A10 holds, that is, the local martingale (cid:18) E (cid:16) q Z · ξ ∗ ( X s ) − ˆ ξ ( X s , s ; T ) dW ,s (cid:17) t (cid:19) ≤ t ≤ T is a true martingale under the measure P .Proof. In order to show this is a true martingale, we use Theorem 8.1 in Klebaner and Liptser (2014). The proof is similar to the proof ofProposition E.1, thus we only state the corresponding functions, a t ( x ) = km − (cid:18) k + qµσ ς + (cid:16) σ + σ − q (cid:17) B (cid:19) x,b t ( x ) = (cid:0) σ √ x, σ √ x (cid:1) ,σ t ( x ) = (cid:16) , q (cid:0) ξ ∗ ( x ) − ˆ ξ ( x, t ; T ) (cid:1)(cid:17) , and it is straightforward to verify that the assumptions of Theorem 8.1 in Klebaner and Liptser (2014) are met.Now the measure Q is well-defined by Eq.(5.17). The functions f and κ in Eq.(5.21) are f ( x, t ; T ) = − qσ x − q ) (cid:0) B − β ( T − t ) (cid:1) (E.3)and κ ( x, t ; T ) = km − (cid:16) k + qµσ ς + σ B (cid:17) x − qσ ˆ ξ ( x, t ; T ) √ x = km − (cid:16) k + qµσ ς + σ B + qσ − q β ( T − t ) (cid:17) x. Finally, the Q -dynamics of X is dX t = κ ( X t , t ; T ) dt + σ p X t dB ,t + σ p X t dB ,t , ≤ t ≤ T. (E.4)37 .1 Integrability condition In the following we prove integrability conditions, which will be needed in the analysis in the next sections.
Lemma E.5.
Under the measure Q , consider two processes U and L defined as the solutions of SDEs dU t = (cid:0) km − υ U U t (cid:1) dt + σ p U t dB t , U = x,dL t = (cid:0) km − υ L L t (cid:1) dt + σ p L t dB t , L = x, where υ U := k + qµσ ς + σ B and υ L := k + qµσ ς + ( σ + σ − q ) B. Then Q (cid:2) L t ≤ X t ≤ U t for all 0 ≤ t ≤ T (cid:3) = 1 . Proof.
Under the measure Q , the process X satisfies dX t = (cid:18) km − (cid:16) υ U + qσ − q β ( T − t ) (cid:17) X t (cid:19) dt + σ p X t dB t , ≤ t ≤ T. Using 0 < β ( · ) < B , we have km − υ L x ≤ km − (cid:16) υ U + qσ − q β ( T − t ) (cid:17) x ≤ km − υ U x. Proposition 5.2.18 in Karatzas and Shreve (1998) gives Q (cid:2) L t ≤ X t ≤ U t for all 0 ≤ t ≤ T (cid:3) = 1 . Lemma E.6.
There are a number u > and an open neighborhood I χ of χ such that Γ u ( x, T ) := E Q h φ u ( X T ) e u R T f ( Xs,s ; T ) ds (cid:12)(cid:12)(cid:12) X = x i is uniformly bounded on I χ × [0 , ∞ ) . Proof.
Since the function f is nonpositive as one can see in Eq.(E.3), it suffices to show that there is a number u > E Q h φ u ( X T ) i = E Q h φ u ( X T ) (cid:12)(cid:12)(cid:12) X = x i is uniformly bounded in ( x, T ) on ( χ , χ ) × [0 , ∞ ) . Recall that the process U in Lemma E.5 satisfies Q (cid:2) X t ≤ U t for all 0 ≤ t ≤ T (cid:3) = 1 . Then for u > E Q h φ u ( X T ) i = E Q [ e uBXT | X = x ] ≤ E Q [ e uBUT | U = x ] . Since U is the a CIR process, it is known that the moment generating function is E Q [ e uBUT , | U = x ] = (cid:16) h T h T − uB (cid:17) kmσ exp (cid:16) uBe − υU T h T xh T − uB (cid:17) where h T = 2 υ U σ (1 − e − υU T ) . Using k + qµσ ς + σ B = υ U , observe that2 B + kσ + qµσ σ ς = υUσ < h T . From this explicit expression, it is easy to check that for1 < u < σ B (cid:16) k + qµσ ς (cid:17) , (E.5)the expectation E Q [ e uBUT | U = x ] is uniformly bounded in ( x, T ) on (cid:0) χ , χ (cid:1) × [0 , ∞ ) . This completes the proof.
E.2 Sensitivity with respect to the initial volatilityProposition E.7.
Under the Heston model, the long-term sensitivity with respect to the initial value of the volatility is lim T →∞ ∂∂χ ln v ( χ, T ) = φ ′ ( χ ) φ ( χ ) = − B. Proof.
By Theorem 3.2, it suffices to prove that the expectation E Q (cid:2) φ ( XT ) e R T f ( Xs,s ; T ) ds (cid:12)(cid:12) X = x (cid:3) is continuously differentiable in x, and ∂∂x E Q h φ ( X T ) e R T f ( Xs,s ; T ) ds (cid:12)(cid:12)(cid:12) X = x i (E.6)converges to zero as T → ∞ . To prove this, we apply Proposition 6.1. Condition (i) of this proposition was proved in Lemma E.6. For (ii),observe that φ ′ ( XT ) φ ( XT ) = − B is a constant, thus this condition holds trivially for any v > .
38e now prove that (iii) holds: for any w > E Q (cid:2) | Y T ; T | w | X = x (cid:3) is uniformly bounded in x on (cid:0) χ , χ (cid:1) and convergesto zero as T → ∞ . From Eq.(E.4), the process Y t = Y t ; T satisfies dY t = − (cid:16) k + qµσ ς + σ B + qσ − q β ( T − t ) (cid:17) Y t dt + σ √ X t Y t dB t , ≤ t ≤ T. By the Itˆo formula, we get X − t Y t = x − e −
12 ( k + qµσ ς + σ B ) t + R t − km + 18 σ
2) 1 Xt − qσ − q β ( T − s )) ds . Since 2 km ≥ σ and β ( · ) ≥ , the integrand in the exponent on the right-hand side is negative. It follows that X − t Y t ≤ x − e −
12 ( k + qµσ ς + σ B ) t . Then for any w > , E Q | Y t | w = E Q (cid:12)(cid:12)(cid:12) X t X − t Y t (cid:12)(cid:12)(cid:12) w ≤ x − w e − w ( k + qµσ ς + σ B ) t E Q h X wt i . (E.7)To obtain (iii), we consider the expectation E Q (cid:2) X wT (cid:3) = E Q (cid:2) X wT (cid:12)(cid:12) X = x (cid:3) . Recall that the process U in Lemma E.5 satisfies Q (cid:2) X t ≤ U t for all 0 ≤ t ≤ T (cid:3) = 1 . Thus, E Q h X wT (cid:12)(cid:12)(cid:12) X = x i ≤ E Q h U wT (cid:12)(cid:12)(cid:12) X = x i . (E.8)On the other hand, since U is a CIR process, for any w > , the expectation on the right hand side is uniformly bounded in ( x, T ) on (cid:0) χ , χ (cid:1) × [0 , ∞ ) . Eq.(E.7) implies that the expectation E Q | Y T | w is uniformly bounded in x on (cid:0) χ , χ (cid:1) and converges to zero as T → ∞ . We now show that (iv) holds for any m > . It is easy to show that there is a positive constant c such that | f x ( x, t ; T ) | = qσ − q ) (cid:12)(cid:12) B − β ( T − t ) (cid:12)(cid:12) ≤ ce − β T − t ) . For convenience, we define δ := ( k + qµσ ς + σ B ) . By Eq.(E.7) and Eq.(E.8), it follows that E Q | Y t ; T | m ≤ b m x − m e − δmt for a positive constant b m which dominates E Q (cid:2) X mt (cid:12)(cid:12) X = x (cid:3) on (cid:0) χ , χ (cid:1) × [0 , ∞ ) . By the Jensen inequality, we have E Q (cid:20)(cid:16)Z T (cid:12)(cid:12) f x ( X s , s ; T ) Y s ; T (cid:12)(cid:12) ds (cid:17) m (cid:21) ≤ c m b m x − m β − δ T m − (cid:0) e − δmT − e − β mT (cid:1) . The right-hand side is uniformly bounded in x on ( χ , χ ) for each T ≥ T → ∞ for each x ∈ (cid:0) χ , χ (cid:1) . This proves(iv). Finally, conditions (ii), (iii), (iv) in Proposition 6.1 hold true for arbitrary v, w, m > , and (i) holds for some u > , so we obtain thedesired result. E.3 Sensitivities with respect to k, m, µ, ς and ρ We calculate the sensitivity with respect to the parameter k. Those with respect to the parameters m, µ, ς and ρ can be calculated in asimilar way. The five functions in B1 and B2 are m ǫ ( x ) = ( k + ǫ )( m − x ) , σ ,ǫ ( x ) = σ √ x, σ ,ǫ ( x ) = σ √ x, b ǫ ( x ) = µx, ς ǫ ( x ) = ς √ x and it is easy to check that they satisfy assumptions B1 and B2. Observe that ∂∂ǫ (cid:12)(cid:12)(cid:12) ǫ =0 ln v ǫ ( χ, T ) = ∂∂k ln v ( χ, T ) = ∂∂k ln v ( χ, T ) , thus for the rest of this section we use ∂∂k instead of ∂∂ǫ | ǫ =0 . Proposition E.8.
Under the Heston model, the long-term sensitivity with respect to the parameter k is lim T →∞ T ∂∂k ln v ( χ, T ) = − ∂λ∂k . Proof.
To prove this equality, we use Theorem 7.1. Condition (i) in Theorem 7.1 is satisfied trivially. We prove (iii) in Theorem 7.1 firstbecause some techniques used for (iii) are also used in the proof of (ii). For condition (iii) in Theorem 7.1, we apply Theorem 7.3. It can beeasily checked that (cid:12)(cid:12)(cid:12) σ √ x ∂∂k κ ( x, t ; T ) (cid:12)(cid:12)(cid:12) ≤ c (cid:16) √ x + 1 √ x (cid:17) , x > c independent of t, T and x > . By choosing sufficiently large c, we can achieve that ˆ g ( x, t ; T ) ≤ c ( x + 1 /x ) holdstrue for ˆ g defined in Eq.(7.4).Then, (i) in Theorem 7.3 can be proven as follows. Recall the processes U and L from Lemma E.5. Since Q [ L t ≤ X t ≤ U t for all 0 ≤ t ≤ T ] = 1 , we have E Q (cid:2) e ǫ R T g Xs,s ; T ) ds (cid:3) ≤ E Q (cid:2) e ǫ c R T Xs +1 /Xs ) ds (cid:3) ≤ E Q (cid:2) e ǫ c R T Us +1 /Ls ) ds (cid:3) ≤ (cid:0) E Q (cid:2) e ǫ c R T Us ds (cid:3)(cid:1) (cid:0) E Q (cid:2) e ǫ c R T /Ls ds (cid:3)(cid:1) . Since U is a CIR process, for given T ≥ ǫ > E Q (cid:2) e ǫ c R T Us ds (cid:3) is finite. In addition, since L is also a CIR processsatisfying the Feller condition, applying Proposition D.2 in Park (2018), one can find ǫ such that E Q (cid:2) e ǫ c R T /Ls ds (cid:3) is finite. This gives(i) in Theorem 7.3. 39ow we prove (ii) in Theorem 7.3 with v = 2 . It suffices to show that there is a positive constant c such that for all T ≥ E Q (cid:20)Z T (cid:16) X s + 1 X s (cid:17) ds (cid:21) ≤ c T. Using the processes U and L in Lemma E.5, observe that E Q (cid:20)Z T (cid:16) X s + 1 X s (cid:17) ds (cid:21) ≤ E Q (cid:20)Z T (cid:16) U s + 1 L s (cid:17) ds (cid:21) = Z T E Q (cid:2) U s (cid:3) + E Q (cid:2) L − s (cid:3) ds. Since U and L are CIR processes satisfying the Feller condition, there is a positive constant c such that for all s ≥ E Q (cid:2) U s (cid:3) + E Q (cid:2) L − s (cid:3) ≤ c . This gives the desired result.For (iii) in Theorem 7.3, we observe that for v = 2 and ǫ = 1 E Q (cid:20)Z T ˆ g v + ǫ ( X s , s ; T ) ds (cid:21) ≤ c / Z T E Q (cid:20)(cid:16) X s + 1 X s (cid:17) / (cid:21) ds ≤ c ′ Z T E Q (cid:2) U / s (cid:3) + E Q (cid:2) L − / s (cid:3) ds. Since U is a CIR process, it is well known that E Q (cid:2) U / s (cid:3) is uniformly bounded in s on [0 , ∞ ) . In addition, for a CIR process L satisfying theFeller condition, we have sup ≤ s ≤ T E Q (cid:2) L − / s (cid:3) < ∞ by Eq.(3.1) in Dereich et al. (2011). This gives (iii) in Theorem 7.3.For (iv) in Theorem 7.3, we want to show that for u = 2 the expectation E Q h φ u ( X T ) e u R T f ( Xs,s ; T ) ds i is uniformly bounded in T on [0 , ∞ ) . We use notation B ( k ) to emphasize the dependence of k on the constant B. From Eq.(E.5), we knowthat for u with 2 < u < σ B ( k + qµσ ς ) the expectation E Q [ e u B ( k ) XT ] is uniformly bounded in T on [0 , ∞ ) . Since the maps k B ( k )is continuous and u > , by choosing a smaller interval I if necessary, it follows thatsup ǫ ∈ I B ( k + ǫ ) ≤ u B ( k ) . Then ˆ φ ( x ) = inf ǫ ∈ I e − B ( k + ǫ ) x ≥ e − u B ( k ) x . (E.9)Thus E Q h φ ( X T ) e R T f ( Xs,s ; T ) ds i ≤ E Q h φ ( X T ) i ≤ E Q (cid:2) e u B ( k ) XT (cid:3) , (E.10)where for the first inequality we used ˆ f ≤ . Since the right-hand side is uniformly bounded in T on [0 , ∞ ) , we obtain the desired result. Wehave now shown all conditions in Theorem 7.3 and thus condition (iii) in Theorem 7.1 holds true. For condition (ii) in Theorem 7.1, we firstcalculate the partial derivative with respect to the variable k in φ and f but not in X = ( X t ) t ≥ . To be precise, we use the notation φ ( x ; k )and f ( x, t ; T ; k ) to emphasize the dependence of k. We want to analyze w η,ǫ ( χ, T ) = E Q ǫ h φ ( X ǫT ; k + η ) e R T f ( Xǫs,s ; T ; k + η ) ds i where the Q ǫ -dynamics of X ǫt satisfies Eq.(E.4) with k replaced by k + ǫ. The equality ∂∂η E Q h φ ( X ǫT ; k + η ) e R T f ( Xǫs,s ; T ; k + η ) ds i = E Q (cid:20) ∂∂η (cid:16) l φ ( X ǫT ; k + η ) e R T f ( Xǫs,s ; T ; k + η ) ds (cid:17)(cid:21) and the continuity of this partial derivative in ( η, ǫ ) on I are obtained from Proposition C.1 with g ( x, t ; T ) and G T given below. Observethat ∂f∂k ( x, t ; T ; k ) = − qσ − q ( B − β ( T − t )) (cid:16) ∂B∂k − ∂β∂k ( T − t ) (cid:17) x. We use the notation β ( T − t ; k ) to emphasize the dependence of k. For a given small open interval I, since B ( k + η ) , ∂B∂k ( k + η ) are continuousin η on I and β ( T − t ; k + η ) , ∂β∂k ( T − t ; k + η ) are continuous in ( η, t ) on I × [0 , T ] , one can find a positive constant b such that for all( η, t ) ∈ I × [0 , T ] (cid:12)(cid:12)(cid:12)(cid:12) ∂f∂η ( x, t ; T ; k + η ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ b x =: g ( t, x ; T ) for x > . With this function g, condition (i) in Proposition C.1 is trivially satisfied.For condition (ii) in Proposition C.1, choose a positive constant b such that for all η ∈ I (cid:12)(cid:12)(cid:12) ∂B∂η ( k + η ) (cid:12)(cid:12)(cid:12) ≤ b . Using the function ˆ φ in Eq.(E.9), we define G T := b X T ˆ φ ( X T ) + 1ˆ φ ( X T ) Z T b X s ds. Then for all ( η, t ) ∈ I × [0 , T ] it follows that1 φ ( X T ; k + η ) (cid:12)(cid:12)(cid:12)(cid:12) ∂φ∂η ( X T ; k + η ) (cid:12)(cid:12)(cid:12)(cid:12) + 1 φ ( X T ; k + η ) Z T (cid:12)(cid:12)(cid:12)(cid:12) ∂f∂η ( X s , s ; T ; k + η ) (cid:12)(cid:12)(cid:12)(cid:12) ds ≤ G T
40y using that ˆ φ ( x ) = inf η ∈ I φ ( x ; k + η ) and (cid:12)(cid:12)(cid:12) ∂φ∂η ( x ; k + η ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) ∂B∂η ( k + η ) (cid:12)(cid:12)(cid:12) xφ ( x ; k + η ) ≤ b xφ ( x ; k + η )for x > . We claim E Q (cid:2) G / T (cid:3) < ∞ , which implies condition (ii) in Proposition C.1. Using that / + = 1, it follows that E Q h φ / ( X T ) (cid:0) b X T (cid:1) / i ≤ (cid:18) E Q h φ ( X T ) i(cid:19) / (cid:18) E Q h(cid:0) b X T (cid:1) i(cid:19) / . The first expectation on the right-hand side is finite by Eq.(E.10). For the second expectation observe that E Q [ X T ] ≤ E Q [ U T ] for the process U in Lemma E.5. Since U is a CIR process, the expectation E Q [ U T ] is finite. In a similar way, we have E Q (cid:20) φ / ( X T ) (cid:16) b Z T X s ds (cid:17) / (cid:21) ≤ (cid:18) E Q h φ ( X T ) i(cid:19) / (cid:18) E Q (cid:20)(cid:16) b Z T X s ds (cid:17) (cid:21)(cid:19) / ≤ b / T / (cid:18) E Q h φ ( X T ) i(cid:19) / (cid:18) E Q (cid:20)(cid:16) T Z T X s ds (cid:17) (cid:21)(cid:19) / ≤ b / T / (cid:18) E Q h φ ( X T ) i(cid:19) / (cid:18) E Q hZ T X s ds i(cid:19) / ≤ b / T / (cid:18) E Q h φ ( X T ) i(cid:19) / (cid:18)Z T E Q [ X s ] ds (cid:19) / . Since E Q [ X s ] ≤ E Q [ U s ] and the expectation E Q [ U s ] is uniformly bounded in s on [0 , T ] , the right-hand side is finite. Hence E Q [ G / T ] < ∞ . The convergence lim T →∞ T ∂∂η (cid:12)(cid:12)(cid:12) η =0 E Q h φ ( X T ; k + η ) e R T f ( Xs,s ; T ; k + η ) ds i = 0can be shown as follows. Using f ≤ , the partial derivative with respect to η satisfies (cid:12)(cid:12)(cid:12)(cid:12) ∂∂η (cid:12)(cid:12)(cid:12) η =0 (cid:16) φ ( X T ; k + η ) e R T f ( Xs,s ; T ; k + η ) ds (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) ≤ e BXT + R T f ( Xs,s ; T ; k ) ds (cid:12)(cid:12)(cid:12) X T ∂B∂k (cid:12)(cid:12)(cid:12) + e BXT + R T f ( Xs,s ; T ; k ) ds (cid:12)(cid:12)(cid:12)(cid:12)Z T ∂f∂η (cid:12)(cid:12)(cid:12) η =0 ( X s , s ; T ; k + η ) ds (cid:12)(cid:12)(cid:12)(cid:12) ≤ e BXT (cid:12)(cid:12)(cid:12) X T ∂B∂k (cid:12)(cid:12)(cid:12) + e BXT (cid:12)(cid:12)(cid:12)(cid:12)Z T ∂f∂η (cid:12)(cid:12)(cid:12) η =0 ( X s , s ; T ; k + η ) ds (cid:12)(cid:12)(cid:12)(cid:12) . By the triangle inequality and the Cauchy–Schwarz inequality, it follows that E Q (cid:12)(cid:12)(cid:12)(cid:12) ∂∂η (cid:12)(cid:12)(cid:12) η =0 (cid:16) φ ( X T ; k + η ) e R T f ( Xs,s ; T ; k + η ) ds (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:16) E Q e BXT (cid:17) / (cid:16) E Q (cid:12)(cid:12)(cid:12) X T ∂B∂k (cid:12)(cid:12)(cid:12) (cid:17) / + (cid:16) E Q e BXT (cid:17) / (cid:18) E Q (cid:12)(cid:12)(cid:12)(cid:12)Z T ∂f∂η (cid:12)(cid:12)(cid:12) η =0 ( X s , s ; T ; k + η ) ds (cid:12)(cid:12)(cid:12)(cid:12) (cid:19) / . By Eq.(E.10), the expectation E Q e BXT is uniformly bounded in T on [0 , ∞ ) . It is easy to show that E Q | X T ∂B∂k | is also uniformly boundedin T on [0 , ∞ ) . Now, we show that the expectation E Q (cid:12)(cid:12)R T ∂f∂η | η =0 ( X s , s ; T ; k + η ) ds (cid:12)(cid:12) is uniformly bounded in T . By direct calculation, onecan choose positive constants c and d, which are independent of s and T but are dependent of k, such that (cid:12)(cid:12)(cid:12)(cid:12) ∂f∂η (cid:12)(cid:12)(cid:12) η =0 ( x, s ; T ; k + η ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ de − c ( T − s ) x. By the same change of variable u = e cs as in Eq.(D.15) and Eq.(D.16), we have that E Q (cid:12)(cid:12)(cid:12)(cid:12)Z T de − c ( T − s ) X s ds (cid:12)(cid:12)(cid:12)(cid:12) is uniformly bounded in T on [0 , ∞ ) . This gives the desired result.
E.4 Sensitivity with respect to σ In this section we calculate the long-term sensitivity with respect to σ. Proposition E.9.
Under the Heston model, the long-term sensitivity with respect to the parameter σ is lim T →∞ T ∂∂σ ln v ( χ, T ) = − ∂λ∂σ . Proof.