Convergence rates of large-time sensitivities with the Hansen--Scheinkman decomposition
aa r X i v : . [ q -f i n . M F ] D ec Convergence rates of large-time sensitivities with theHansen–Scheinkman decomposition
Hyungbin Park ∗ Department of Mathematical SciencesSeoul National University1, Gwanak-ro, Gwanak-gu, Seoul, Republic of Korea
December 10, 2019
Abstract
This paper investigates the large-time asymptotic behavior of the sensitivities of cash flows.In quantitative finance, the price of a cash flow is expressed in terms of a pricing operatorof a Markov diffusion process. We study the extent to which the pricing operator is affectedby small changes of the underlying Markov diffusion. The main idea is a partial differentialequation (PDE) representation of the pricing operator by incorporating the Hansen–Scheinkmandecomposition method. The sensitivities of the cash flows and their large-time convergencerates can be represented via simple expressions in terms of eigenvalues and eigenfunctions ofthe pricing operator. Furthermore, compared to the work of Park (Finance Stoch. 4:773-825,2018), more detailed convergence rates are provided. In addition, we discuss the applicationof our results to three practical problems: utility maximization, entropic risk measures, andbond prices. Finally, as examples, explicit results for several market models such as the Cox–Ingersoll–Ross (CIR) model, 3/2 model and constant elasticity of variance (CEV) model arepresented.
In financial mathematics, sensitivity analysis is used to demonstrate how changes of parametersaffect cash flows. A cash flow is expressed in expectation form as p T := E P ξ [ e − R T r ( X s ) ds h ( X T )] , (1.1)where E P ξ is an expectation, r and h are suitable measurable functions, and X = ( X t ) t ≥ isan underlying stochastic process with X = ξ. This paper deals with the sensitivities of theexpectation p T with respect to changes of the underlying process X as well as their large-timeasymptotic behaviors as T → ∞ . The underlying process X is assumed to be a Markov diffusion,and the Markov process X with killing rate r generates a pricing operator. The main idea is apartial differential equation (PDE) representation of the pricing operator by incorporating the ∗ [email protected], [email protected] ansen–Scheinkman decomposition method. We conclude that the large-time behavior of thesensitivities is expressed in terms of eigenvalues and eigenfunctions of the pricing operator.One of the core concepts of this paper is the Hansen–Scheinkman decomposition. Undersuitable conditions, the expectation p T is expressed as p T = φ ( ξ ) e − λT f ( T, ξ ) (1.2)for a positive measurable function φ ( · ) , a positive number λ , and a measurable function f ( T, · ) . The function f ( T, ξ ) converges to a nonzero constant, which is independent of ξ , as T → ∞ . The key aspect of this decomposition is that the function f is regarded as a negligible term as T → ∞ so that the behavior of p T is determined by the two factors φ ( ξ ) and e − λT . In particular,the large-time behavior of p T satisfies | p T | ≤ ce − λT for a positive constant c, independent of T. This paper essentially investigates two types of sensitivities. The first is the sensitivity withrespect to the initial value X = ξ, which will be discussed in Section 3. For the first-ordersensitivity, known as the delta value, the derivative ∂ ξ p T and its large-time behavior as T → ∞ are of interest to us. From the Hansen–Scheinkman decomposition presented in Eq.(1.2), itfollows that ∂ ξ p T p T = φ ′ ( ξ ) φ ( ξ ) + f x ( T, ξ ) f ( T, ξ ) . We show that the derivative f x ( T, ξ ) can also be expressed in expectation form in Eq.(1.1),and we then apply the Hansen–Scheinkman decomposition repeatedly. Similar to Eq.(1.2), thederivative f x ( T, ξ ) has the decomposition f x ( T, ξ ) = ˆ φ ( ξ ) e − ˆ λT ˆ f ( T, ξ )for a positive measurable function ˆ φ ( · ) , a positive number ˆ λ , and a measurable function ˆ f ( T, · )converging to a nonzero constant, which is independent of ξ , as T → ∞ . Finally, we have (cid:12)(cid:12)(cid:12)(cid:12) ∂ ξ p T p T − φ ′ ( ξ ) φ ( ξ ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) f x ( T, ξ ) f ( T, ξ ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ ce − ˆ λT , T ≥ c. Thus, ∂ ξ p T p T converges to φ ′ ( ξ ) φ ( ξ ) as T → ∞ and its exponentialconvergence rate is ˆ λ. Further details are discussed in Theorem 3.1. For the second-order sensi-tivity, known as the gamma value, we present a similar argument for ∂ ξξ p T and its asymptoticbehavior. We show that (cid:12)(cid:12)(cid:12)(cid:12) ∂ ξξ p T p T − φ ′′ ( ξ ) φ ( ξ ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ ce − ˆ λT for some positive constant c. The asymptotic behavior is discussed in further detail in Theorem3.2.The second type of sensitivity includes the drift and diffusion sensitivities, which are knownas the rho value and the vega value, respectively. Let ( X ( ǫ ) t ) t ≥ be the underlying process withperturbed drift or diffusion terms. The precise meaning of perturbation is given in Assumption3.1. Here, ǫ can be understood as a perturbation parameter. Let p ( ǫ ) T := E P ξ [ e − R T r ( X ǫs ) ds f ( X ǫT )]be the expectation corresponding to the perturbed underlying process. We want to investigatethe large-time behavior of ∂∂ǫ (cid:12)(cid:12)(cid:12) ǫ =0 p ( ǫ ) T . Under the assumption that the Hansen–Scheinkman decomposition is applicable to each ǫ (As-sumption 3.2), we obtain p ( ǫ ) T = φ ( ǫ ) ( ξ ) e − λ ( ǫ ) T f ( ǫ ) ( T, ξ ) , hich is analogous to Eq.(1.2). We verify that under some circumstances, (cid:12)(cid:12)(cid:12)(cid:12) T ∂∂ǫ (cid:12)(cid:12)(cid:12) ǫ =0 ln p ( ǫ ) T + ∂∂ǫ (cid:12)(cid:12)(cid:12) ǫ =0 λ ( ǫ ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ cT for a positive constant c. Thus, T ∂∂ǫ | ǫ =0 ln p ( ǫ ) T converges to − ∂∂ǫ | ǫ =0 λ ( ǫ ) as T → ∞ and itsconvergence rate is O (1 /T ) . Compared to the previous work of Park (2018), this paper has three distinguishing features.First, for the first-order sensitivity with respect to the initial value X = ξ , the exponentialconvergence rate is demonstrated. Eq.(1.3) (or Theorem 3.1) implies that ∂ ξ p T p T converges to φ ′ ( ξ ) φ ( ξ ) as T → ∞ and its exponential convergence rate is ˆ λ. The previous work also showed that ∂ ξ p T p T converges to φ ′ ( ξ ) φ ( ξ ) as T → ∞ ; however, its exponential convergence rate was not provided.Second, the second-order sensitivity with respect to the initial value X = ξ is analyzed in thispaper (Section 3.1.2), whereas it was not addressed in the previous work at all. Third, for thedrift and diffusion perturbations, the current paper adopts more relaxed assumptions comparedto the previous work. As the previous work relies on Malliavin calculus, it requires strongconditions such as continuous differentiability with bounded derivatives and uniform ellipticityon drift and diffusion functions. As a specific example, if the underlying process is the constantelasticity of variance (CEV) model dX t X t = µ dt + σX tβ dB t , X = ξ , this paper can address the sensitivity with respect to the leverage effect parameter β (Sections5.3 and 5.4), whereas it cannot be analyzed by the method used in the previous work.Sensitivity analysis has been studied for various topics in quantitative finance. Fourni´e et al.(1999) presented an original probabilistic method for numerical computation of the sensitivities.They employed Malliavin calculus and demonstrated option price sensitivities for hedging pur-poses. Gobet and Munos (2005) derived an expectation form of the sensitivity of the expectedcost by employing three methods: the Malliavin calculus approach, the adjoint approach, andthe martingale approach. Kramkov and Sˆırbu (2006) developed sensitivity analysis of the op-timal expected utility with respect to initial capital perturbations. Mostovyi and Sˆırbu (2017)and Mostovyi (2018) conducted sensitivity analysis of the optimal expected utility with respectto small changes of the underlying market models. Park and Sturm (2019) investigated thesensitivities of the long-term expected utility of optimal portfolios for an investor with constantrelative risk aversion under incomplete markets.Many authors have investigated the behavior of long-term cash flows. Fleming and McEneaney(1995) studied the long-term growth of expected utility with constant relative risk aversion andreformulated it as an infinite-time-horizon risk-sensitive control problem. Liu and Muhle-Karbe(2013) demonstrated a computational method for evaluating optimal portfolios with special em-phasis on long-horizon asymptotics. Robertson and Xing (2015) analyzed the large-time asymp-totic behavior of solutions to semi-linear Cauchy problems with direct applications to long-termportfolio choice problems. Hansen (2012), Hansen and Scheinkman (2009), and Hansen and Scheinkman(2012) demonstrated a long-term risk-return trade-off by employing the Hansen–Scheinkman de-composition.The remainder of this paper is organized as follows. Section 2 introduces the Hansen–Scheinkman decomposition method as an essential tool of this paper. Section 3 investigatesthe sensitivities with respect to the initial value, the drift term, and the diffusion term anddemonstrates their large-time behaviors. Section 4 discusses direct applications to three topics:utility maximization, entropic risk measures, and bond prices. Section 5 presents three specificexamples: the Cox–Ingersoll–Ross (CIR) model, the 3/2 model, and the CEV model. Finally,Section 6 summarizes the results. The proofs and detailed calculations are provided in theappendices. Hansen–Scheinkman decomposition
The Hansen–Scheinkman decomposition (Hansen and Scheinkman (2009)) is one of the maintechniques employed in this paper. Given a Markov diffusion ( X t ) t ≥ and a function r ( · ) , thisdecomposition provides an expression of the operator h E P ξ [ e − R T r ( X s ) ds h ( X T )]in terms of eigenvalues and eigenfunctions of this operator. In the following sections, we willdescribe its mathematical formulation. We begin with the notion of a consistent family of probability measures. While considering afiltered probability space (Ω , F , ( F t ) t ≥ , P ) , the probability measure P is an object defined onthe sigma-algebra F . The probability measure P is universal in the sense that the sigma-algebracontains all sub-sigma-algebras ( F t ) t ≥ . Instead of such a universal probability measure, weintroduce a family of probability measures ( P t ) t ≥ where each P t is defined on the sub-sigma-algebra F t .The main reason for introducing this concept is that defining such a universal probabilitymeasure is occasionally impossible. As a specific case, the change of measure in the Girsanovtheorem holds for finite time horizon [0 , T ] but it may not hold for infinite time horizon [0 , ∞ ).Thus, to use the Girsanov theorem for studying large-time behavior as T → ∞ , it is moreconvenient to deal with a family of probability measures instead of a universal probabilitymeasure. Definition 2.1.
Let (Ω , F ) be a measurable space and ( F t ) t ≥ be a filtration. We say that afamily ( P t ) t ≥ of probability measures is consistent if each probability measure P t is defined on F t and if P t ( A ) = P t ′ ( A ) for any ≤ t ≤ t ′ and any A ∈ F t . We say that (Ω , F , ( F t ) t ≥ , ( P t ) t ≥ ) is a consistent probabilityspace . In this paper, we abuse the notations P and P t without ambiguity. For a F t -measurable randomvariable X, the expectation E P t ( X ) is denoted as E P ( X ) . This notation is not confusing becausethe family ( P t ) t ≥ is consistent so that E P t ( X ) = E P t ′ ( X ) for any t ′ ≥ t. We present basic definitions of several probabilistic concepts. These definitions are straight-forward.
Definition 2.2.
Let (Ω , F , ( F t ) t ≥ , ( P t ) t ≥ ) be a consistent probability space.(i) We say that a process B = ( B t ) t ≥ is a d -dimensional Brownian motion on the consistentprobability space if for each T ≥ , the process ( B t ) ≤ t ≤ T is a usual d -dimensional Brownianmotion on the filtered probability space (Ω , F T , ( F t ) ≤ t ≤ T , P T ) . (ii) We say that a process X = ( X t ) t ≥ is a Markov process (respectively, a martingale) on theconsistent probability space if for each T ≥ , the process ( X t ) ≤ t ≤ T is a Markov process(respectively, a martingale) on the filtered probability space (Ω , F T , ( F t ) ≤ t ≤ T , P T ) . (iii) We say that a process X = ( X t ) t ≥ is a unique strong solution of the SDE (respectively,satisfies the SDE) dX t = b ( X t ) dt + σ ( X t ) dB t , X = ξ on the consistent probability space if for each T ≥ , the process ( X t ) ≤ t ≤ T is a uniquestrong solution of the SDE (respectively, satisfies the SDE) on the filtered probability space (Ω , F T , ( F t ) ≤ t ≤ T , P T ) . iv) Let D be an open subset of R d and X be a Markov process with state space D on theconsistent probability space. We say that the process X is recurrent if R ( x, A ) := Z ∞ E P ( I A ( X t )) dt = 0 or R ( x, A ) = ∞ for any Borel set A ⊆ D and any x ∈ D . Refer to (Pinsky, 1995, Theorem 4.3.6 on page150) and (Qin and Linetsky, 2016, Definition 3.1).
Let (Ω , F , ( F t ) t ≥ , ( P t ) t ≥ ) be a consistent probability space that has a one-dimensional Brow-nian motion ( B t ) t ≥ . We consider a quadruple of functions ( b, σ, r, h ) satisfying Assumptions2.1–2.5 below on the consistent probability space (Ω , F , ( F t ) t ≥ , ( P t ) t ≥ ) with given Brownianmotion ( B t ) t ≥ . Assumption 2.1.
Let D be an open interval in R and b : D → R and σ : D → R be continuouslydifferentiable functions with σ > . For each ξ ∈ D , the SDE dX t = b ( X t ) dt + σ ( X t ) dB t , X = ξ has a unique strong solution on D , i.e., P T ( X t ∈ D for all 0 ≤ t ≤ T ) = 1 for each T ≥ . Assumption 2.2.
The function r : D → R is a continuous function. For given b, σ and r satisfying the above-mentioned assumptions, we can define a pricingoperator P as P T h ( x ) = E P x ( e − R T r ( X s ) ds h ( X T )) , x ∈ D so that p T = P T h ( ξ ) . For a real number λ and a positive measurable function φ, we say that apair ( λ, φ ) is an eigenpair of P if P T φ ( x ) = e − λT φ ( x ) for all T > , x ∈ D . For each eigenpair ( λ, φ ) , the process M φt := e λt − R t r ( X s ) ds φ ( X t ) φ ( ξ ) , t ≥ P φt on each F t by d ˆ P φt d P t = M φt . Then, it can be easily checked that the family (ˆ P φt ) t ≥ is consistent, i.e.,ˆ P φt ( A ) = ˆ P φt ′ ( A )for any A ∈ F t and 0 ≤ t ≤ t ′ . The family (ˆ P φt ) t ≥ is called the consistent family of the eigen-measures with respect to φ. Using the Girsanov theorem, we can show that the processˆ B φt = − Z t ( σφ ′ /φ )( X s ) ds + B t , ≤ t ≤ T is a Brownian motion on the filtered probability space (Ω , F T , ( F t ) ≤ t ≤ T , ˆ P φT ) for each T ≥ . In other words, the process ( ˆ B φt ) t ≥ is a Brownian motion on the consistent probability space Ω , F , ( F t ) t ≥ , (ˆ P φt ) t ≥ ) (see Definition 2.2). Similarly, it is easy to check that the process X satisfies dX t = ( b + σ φ ′ /φ )( X t ) dt + σ ( X t ) d ˆ B φt on the consistent probability space (Ω , F , ( F t ) t ≥ , (ˆ P φt ) t ≥ ) . Assumption 2.3.
There exists an eigenpair ( λ, φ ) of the operator P with λ > such that theprocess X is recurrent on the consistent probability space (Ω , F , ( F t ) t ≥ , (ˆ P φt ) t ≥ ) . In this case, the positive number λ , the measurable function φ , and the pair ( λ, φ ) are called the recurrent eigenvalue , the recurrent eigenfunction , and the recurrent eigenpair , respectively. Sev-eral studies have investigated the existence of the recurrent eigenpair, e.g., (Hansen and Scheinkman,2009, Section 9) and (Qin and Linetsky, 2016, Section 5). Hereafter, we use the notations M, ˆ P , (ˆ P t ) t ≥ , and ( ˆ B t ) t ≥ instead of M φ , ˆ P φ , (ˆ P φt ) t ≥ , and ( ˆ B φt ) t ≥ , respectively, without φ. Thesenotations are not confusing because the recurrent eigenpair ( λ, φ ) is unique if it exists (Proposi-tion 7.2 in Hansen and Scheinkman (2009) and Theorem 3.1 in Qin and Linetsky (2016)). It isalso noteworthy that when the recurrent eigenpair exists, if r ≥ r
0, then λ > . Referto (Pinsky, 1995, Theorem 3.3 (iii) on page 148).
Assumption 2.4.
The recurrent eigenfunction φ is twice continuously differentiable on D . A two-variable function f = f ( t, x ) is said to be C , if f is once continuously differentiablewith respect to t and twice continuously differentiable with respect to x. Assumption 2.5.
The function h : D → R is continuously differentiable. The function f ( t, x ) := E P x [ M t ( h/φ )( X t )] = E ˆ P x [( h/φ )( X t )] (2.2) is C , and converges to a nonzero constant as t → ∞ for each x ∈ D . This function f is referredto as the remainder function . For a given quadruple of functions ( b, σ, r, h ) satisfying Assumptions 2.1–2.5 on the consistentprobability space (Ω , F , ( F t ) t ≥ , ( P t ) t ≥ ) with Brownian motion ( B t ) t ≥ , we have constructedthe process X, the pricing operator P , the recurrent eigenpair ( λ, φ ) , the martingale M, the re-mainder function f, the consistent family (ˆ P t ) t ≥ of recurrent eigen-measures, and the Brownianmotion ( ˆ B t ) t ≥ . Hereafter, these objects X, P , ( λ, φ ) , M, f, (ˆ P t ) t ≥ , ( ˆ B t ) t ≥ appear frequently and the notations are self-explanatory.Under these assumptions, the Hansen–Scheinkman decomposition is a very useful tool forlarge-time analysis. From Eq.(2.1), the discount factor e − R T r ( X t ) dt can be written as e − R T r ( X s ) ds = M T e − λT φ ( ξ ) φ ( X T ) . This expression is referred to as the Hansen–Scheinkman decomposition. The expectation p T satisfies p T = E P ξ [ e − R T r ( X s ) ds h ( X T )] = φ ( ξ ) e − λT E P ξ [ M T ( h/φ )( X T )]= φ ( ξ ) e − λT E ˆ P ξ [( h/φ )( X T )]= φ ( ξ ) e − λT f ( T, ξ ) . (2.3)Because f ( T, ξ ) converges to a nonzero constant as T → ∞ , we obtain the inequality | p T | ≤ ce − λT for some positive constant c, which is independent of T. This implies that the large-time behaviorof p T is governed by the recurrent eigenvalue. Long-term sensitivity analysis
This section develops the Hansen–Scheinkman decomposition presented above to investigate thelarge-time behavior of the sensitivities.
We begin with the initial-value sensitivity, i.e., the extent to which the expectation p T := E P ξ [ e − R T r ( X s ) ds h ( X T )]is affected by small changes of the initial value ξ = X of the underlying Markov diffusion.The first-order and second-order sensitivities with respect to the initial value are demonstratedbelow. In this section, we develop the first-order initial-value sensitivity for large time T. For a givenquadruple of functions ( b, σ, r, h ) satisfying Assumptions 2.1–2.5, the large-time asymptotic be-havior of the partial derivative ∂ ξ p T = ∂ ξ E P ξ [ e − R T r ( X s ) ds h ( X T )]is of interest to us. For notational simplicity, we define κ := b + σ φ ′ /φ . If ( κ + σ ′ σ, σ, − κ ′ , ( h/φ ) ′ ) also satisfies Assumptions 2.1–2.5 on the consistent probability space(Ω , F , ( F t ) t ≥ , (ˆ P t ) t ≥ ) with Brownian motion ( ˆ B t ) t ≥ , we can construct the corresponding ob-jects ˆ X, ˆ P , (ˆ λ, ˆ φ ) , ˆ M , ˆ f , (˜ P t ) t ≥ , ( ˜ B t ) t ≥ and these notations are self-explanatory. For example, ˆ X satisfies d ˆ X t = ( κ + σ ′ σ )( ˆ X t ) dt + σ ( ˆ X t ) d ˆ B t . (3.1)These objects will be used in the statement and the proof of the following theorem. Theorem 3.1.
Suppose that the following conditions hold.(i) The quadruple of functions ( b, σ, r, h ) satisfies Assumptions 2.1–2.5 on the consistent prob-ability space (Ω , F , ( F t ) t ≥ , ( P t ) t ≥ ) with Brownian motion ( B t ) t ≥ . (ii) The quadruple of functions ( κ + σ ′ σ, σ, − κ ′ , ( h/φ ) ′ ) satisfies Assumptions 2.1–2.5 on theconsistent probability space (Ω , F , ( F t ) t ≥ , (ˆ P t ) t ≥ ) with Brownian motion ( ˆ B t ) t ≥ . Then, for each
T > , the process ( f x ( T − t, ˆ X t ) e R t κ ′ ( ˆ X s ) ds ) ≤ t ≤ T is a local martingale underthe probability measure ˆ P T . For each
T > , if this process is a martingale under the probabilitymeasure ˆ P T or if the function f x satisfies f x ( T, x ) = E ˆ P [ e R T κ ′ ( ˆ X s ) ds ( h/φ ) ′ ( ˆ X T ) | ˆ X = x ] , (3.2) then (cid:12)(cid:12)(cid:12)(cid:12) ∂ ξ p T p T − φ ′ ( ξ ) φ ( ξ ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ ce − ˆ λT , T ≥ for some positive constant c, which is independent of T. roof. Applying the Feynman–Kac formula to Eq.(2.2), it follows that − f t + 12 σ ( x ) f xx + κ ( x ) f x = 0 , f (0 , x ) = ( h/φ )( x ) . Since f is C , by Assumption 2.5 and every coefficient is continuously differentiable in x in theabove PDE, the function f is thrice continuously differentiable in x. Taking the differentiationin x, we get − f xt + 12 σ ( x ) f xxx + ( κ + σ ′ σ )( x ) f xx + κ ′ ( x ) f x = 0 , f x (0 , x ) = ( h/φ ) ′ ( x ) . (3.4)Meanwhile, since ( κ + σ ′ σ, σ, − κ ′ , ( h/φ ) ′ ) satisfies Assumptions 2.1–2.5 on the consistentprobability space (Ω , F , ( F t ) t ≥ , (ˆ P t ) t ≥ ) having Brownian motion ( ˆ B t ) t ≥ , we can construct thecorresponding objects ˆ X, ˆ P , (ˆ λ, ˆ φ ) , ˆ M , ˆ f , (˜ P t ) t ≥ , ( ˜ B t ) t ≥ and these notations are self-explanatory. Since the process ˆ X satisfies d ˆ X t = ( κ + σ ′ σ )( ˆ X t ) dt + σ ( ˆ X t ) d ˆ B t , by applying the Ito formula to f x ( T − t, ˆ X t ) e R t κ ′ ( X s ) ds , we have d (cid:0) f x ( T − t, ˆ X t ) e R t κ ′ ( ˆ X s ) ds (cid:1) = e R t κ ′ ( ˆ X s ) ds (cid:16) − f xt + 12 σ ( ˆ X t ) f xxx + ( κ + σ ′ σ )( ˆ X t ) f xx + κ ′ ( ˆ X s ) f x (cid:17) dt + e R t κ ′ ( ˆ X s ) ds σ ( ˆ X t ) f xx d ˆ B t = e R t κ ′ ( ˆ X s ) ds σ ( ˆ X t ) f xx d ˆ B t . (3.5)Thus, for each T >
0, the process ( f x ( T − t, ˆ X t ) e R t κ ′ ( ˆ X s ) ds ) ≤ t ≤ T is a local martingale underthe probability measure ˆ P T . Suppose that f x satisfies f x ( T, x ) = E ˆ P [ e R T κ ′ ( ˆ X s ) ds ( h/φ ) ′ ( ˆ X T ) | ˆ X = x ] . Note that this equality holds if ( f x ( T − t, ˆ X t ) e R t κ ′ ( ˆ X s ) ds ) ≤ t ≤ T is a martingale since f x ( T, x ) = E ˆ P [ e R T κ ′ ( ˆ X s ) ds f x (0 , ˆ X T ) | ˆ X = x ] = E ˆ P [ e R T κ ′ ( ˆ X s ) ds ( h/φ ) ′ ( ˆ X T ) | ˆ X = x ] . (3.6)Replacing T by t, it follows that f x ( t, x ) = E ˆ P (cid:2) e R t κ ′ ( ˆ X s ) ds ( h/φ ) ′ ( ˆ X t ) (cid:12)(cid:12) ˆ X = x (cid:3) = ˆ P t ( h/φ ) ′ ( x ) . Now, we apply the Hansen–Scheinkman decomposition here. Since (ˆ λ, ˆ φ ) is the recurrent eigen-pair and ˆ f ( t, x ) = E ˆ P x [ ˆ M t ( h/φ ) ′ ( ˆ X t )] = E ˜ P x [(( h/φ ) ′ / ˆ φ )( ˆ X t )]is the remainder function, we have f x ( t, x ) = ˆ f ( t, x ) e − ˆ λt ˆ φ ( x ) . (3.7)Since f ( t, x ) and ˆ f ( t, x ) converge to nonzero constants as t → ∞ , Eq.(2.3) implies that (cid:12)(cid:12)(cid:12)(cid:12) ∂ ξ p T p T − φ ′ ( ξ ) φ ( ξ ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) f x ( T, ξ ) f ( T, ξ ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ f ( T, ξ ) f ( T, ξ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e − ˆ λT ˆ φ ( x ) ≤ ce − ˆ λT (3.8)for some positive constant c, which is independent of T. emark 3.1. Theorem 3.1 has an interesting implication. Eq. (3.3) says lim T →∞ ∂ ξ p T p T = φ ′ ( ξ ) φ ( ξ ) . This means that the large-time behavior of the sensitivity ∂ ξ p T p T as T → ∞ is expressed in terms ofthe recurrent eigenfunction φ induced by ( b, σ, r, h ) . However, as one can observe from Eq. (3.3) ,its exponential convergence rate to the limit is determined by the recurrent eigenvalue ˆ λ inducedby ( κ + σ ′ σ, σ, − κ ′ , ( h/φ ) ′ ) . Remark 3.2.
Under the same hypothesis of Theorem 3.1, the probabilistic representation f x ( T, x ) = E ˆ P [ e R T κ ′ ( ˆ X s ) ds ( h/φ ) ′ ( ˆ X T ) | ˆ X = x ] stated in Eq. (3.2) holds if E ˆ P ξ h Z T e R t κ ′ ( ˆ X s ) ds σ ( ˆ X t ) f xx ( T − t, ˆ X t ) dt i < ∞ . This is evident from Eq. (3.5) since the process ( f x ( T − t, ˆ X t ) e R t κ ′ ( ˆ X s ) ds ) ≤ t ≤ T is a martingaleunder the probability measure ˆ P T . Recall the hypothesis in Theorem 3.1: the process ( f x ( T − t, ˆ X t ) e R t κ ′ ( ˆ X s ) ds ) ≤ t ≤ T is a P T -martingale or the function f x satisfies Eq.(3.2). It can be easily shown that these two statementsare equivalent. In the remainder of this section, we investigate a sufficient condition to guaranteethat this hypothesis holds, i.e., the function f x satisfies f x ( T, x ) = E ˆ P [ e R T κ ′ ( ˆ X s ) ds ( h/φ ) ′ ( ˆ X T ) | ˆ X = x ]presented in Eq.(3.2). The sufficient condition is based on the Feynman–Kac formula and isgiven in Proposition 3.1.Before stating Proposition 3.1, we present a slight modification of the standard Feynman–Kac formula (Karatzas and Shreve, 1991, Theorem 5.7.6 on page 366) in the remark below.Recall that D is an open interval in R . We say that a function f on D has polynomial growth(respectively, linear growth) if there is a constant C > m ∈ N (respectively, m = 1) suchthat for all x ∈ D , | f ( x ) | ≤ C (1 + | x | m ) . Remark 3.3. (Feynman–Kac formula) Consider a quadruple of functions ( b, σ, r, h ) defined onan open interval D ⊆ R . Suppose that the following conditions hold.(i) Two functions b and σ are continuous and have linear growth.(ii) For each x ∈ D , the stochastic differential equation (SDE) dX t = b ( X t ) dt + σ ( X t ) dB t , X = x has a unique strong solution on D , i.e., P T ( X t ∈ D for all 0 ≤ t ≤ T ) = 1 for each T ≥ . (iii) Three functions r, h and g are continuous. In addition, r is bounded below, h has poly-nomial growth or is nonnegative, and max ≤ t ≤ T | g ( t, · ) | has polynomial growth or g isnonnegative.If f ( t, x ) is C , and satisfies − f t ( t, x ) + 12 σ ( x ) f xx ( t, x ) + b ( x ) f x ( t, x ) − r ( x ) f ( t, x ) + g ( t, x ) = 0 , f (0 , x ) = h ( x ) (3.9) s well as the polynomial growth condition max ≤ t ≤ T | f ( t, x ) | ≤ C (1 + | x | m ) for some C > and m ≥ , then f ( t, x ) = E h e − R t r ( X s ) ds h ( X t ) + Z t g ( T − s, X s ) e − R s r ( X u ) du ds (cid:12)(cid:12)(cid:12) X = x i (3.10) for x ∈ D and ≤ t ≤ T. The Feynman–Kac formula stated above differs from the standard statement (Karatzas and Shreve,1991, Theorem 5.7.6 on page 366) in two ways. First, the domain in this case is an open interval D , whereas the domain is the whole real line R in the case of the standard statement. Thisis easily verified because the proof of the standard statement can be directly applied. Second,the time-derivative term − f t in the PDE (3.9) has a negative sign. The PDE in the standardstatement is expressed in the time-reverse order, i.e., the final-time condition f ( T, · ) is given.However, in this paper, we are interested in the time order, i.e., the initial-time condition f (0 , · )is given. The time-reverse order can be easily changed to the time order by using the Markovproperty. For fixed T > , define F ( t, x ) = f ( T − t, x ) . Then, Eq.(3.9) becomes F t ( t, x ) + 12 σ ( x ) F xx ( t, x ) + b ( x ) F x ( t, x ) − r ( x ) F ( t, x ) + g ( T − t, x ) = 0 , F ( T, x ) = h ( x ) . From the standard Feynman–Kac formula, we know that F ( t, x ) = E h e − R Tt r ( X s ) ds h ( X T ) + Z Tt g ( T − s, X s ) e − R st r ( X u ) du ds (cid:12)(cid:12)(cid:12) X t = x i . Thus, f ( T, x ) = F (0 , x ) = E h e − R T r ( X s ) ds h ( X T ) + Z T g ( T − s, X s ) e − R s r ( X u ) du ds (cid:12)(cid:12)(cid:12) X = x i . Replacing T by t, we obtain Eq.(3.10).The Feynman–Kac formula in the above-mentioned remark occasionally cannot be appliedto obtain the representation f x ( T, x ) = E ˆ P [ e R T κ ′ ( ˆ X s ) ds ( h/φ ) ′ ( ˆ X T ) | ˆ X = x ]because it requires the function max ≤ t ≤ T | f x ( t, x ) | to have polynomial growth in x. Some fi-nancial models do not satisfy this condition (e.g., the CIR model in Appendix A). One wayto overcome this problem is to consider f x ( t, x ) φ ( x ) instead of f x ( t, x ) . The main aspect ofProposition 3.1 is that the polynomial growth condition on max ≤ t ≤ T | f x ( t, x ) | can be replacedby the polynomial growth condition on max ≤ t ≤ T | f x ( t, x ) | φ ( x ) . Indeed, in the CIR model, thefunction max ≤ t ≤ T | f x ( t, x ) | φ ( x ) has polynomial growth in x, whereas max ≤ t ≤ T | f x ( t, x ) | doesnot. Proposition 3.1.
Suppose that the following conditions hold.(i) The quadruple of functions ( b, σ, r, h ) satisfies Assumptions 2.1–2.5 on the consistent prob-ability space (Ω , F , ( F t ) t ≥ , ( P t ) t ≥ ) with Brownian motion ( B t ) t ≥ . (ii) The quadruple of functions ( κ + σ ′ σ, σ, − κ ′ , ( h/φ ) ′ ) satisfies Assumptions 2.1–2.5 on theconsistent probability space (Ω , F , ( F t ) t ≥ , (ˆ P t ) t ≥ ) with Brownian motion ( ˆ B t ) t ≥ . Furthermore, assume the following conditions for given
T > . i) Two functions κ + σ ′ σ − σ φ ′ /φ and σ have linear growth.(ii) The function ( ˆ L φ ) /φ − κ ′ is bounded below, where ˆ L f ( x ) = 12 σ ( x ) f ′′ ( x ) + ( κ + σ ′ σ − σ φ ′ /φ )( x ) f ′ ( x ) . (iii) The function h ′ − hφ ′ /φ has polynomial growth or is nonnegative.(iv) The function max ≤ t ≤ T | f x ( t, x ) | φ ( x ) has polynomial growth in x. (v) A local martingale φ ( ˆ X ) φ ( ˆ X t ) e R t L φ ( ˆ Xs ) φ ( ˆ Xs ) ds , ≤ t ≤ T is a martingale under the probability measure ˆ P T . Then, the process ( f x ( T − t, ˆ X t ) e R t κ ′ ( ˆ X s ) ds ) ≤ t ≤ T is a martingale under the probability measure ˆ P T . In particular, f x ( T, x ) = E ˆ P [ e R T κ ′ ( ˆ X s ) ds ( h/φ ) ′ ( ˆ X T ) | ˆ X = x ] . Proof.
Recall that (Ω , F , ( F t ) t ≥ , (ˆ P t ) t ≥ ) is a consistent probability space having Brownianmotion ( ˆ B t ) t ≥ and the process ( ˆ X t ) t ≥ satisfies Eq.(3.1). One can show that φ ( ˆ X ) φ ( ˆ X t ) e R t L φ ( ˆ Xs ) φ ( ˆ Xs ) ds , ≤ t ≤ T is a ˆ P T -local martingale by applying the Ito formula and checking that the dt -term vanishes.Since this process is assumed to be a martingale, we can define a new measure Q T on F T as d Q T d ˆ P T = φ ( ˆ X ) φ ( ˆ X T ) e R T L φ ( ˆ Xs ) φ ( ˆ Xs ) ds . It is easy to check that the family ( Q t ) t> of probability measures is consistent and the process B Q t := ˆ B t + ( σφ ′ /φ )( ˆ X t ) , t ≥ , F , ( F t ) t ≥ , ( Q t ) t ≥ ) . The processˆ X satisfies d ˆ X t = ( κ + σ ′ σ )( ˆ X t ) dt + σ ( ˆ X t ) d ˆ B t = ( κ + σ ′ σ − σ φ ′ /φ )( ˆ X t ) dt + σ ( ˆ X t ) dB Q t . Note that ˆ L is the generator of ˆ X under the family of probability measures ( Q t ) t ≥ . Define g ( t, x ) := f x ( t, x ) φ ( x ) . Then, Eq.(3.4) gives − g t + 12 σ ( x ) g xx + ( κ + σ ′ σ − σ φ ′ /φ )( x ) g x + ( κ ′ − ( ˆ L φ ) /φ )( x ) g = 0 g (0 , x ) = ( h/φ ) ′ ( x ) φ ( x ) = ( h ′ − hφ ′ /φ )( x ) . The Feynman–Kac formula (Remark 3.3) states that f x ( t, x ) φ ( x ) = g ( t, x ) = E Q (cid:2) e R t ( κ ′ − ˆ L φφ )( ˆ X s ) ds φ ( ˆ X t )( h/φ ) ′ ( ˆ X t ) (cid:12)(cid:12) ˆ X = x (cid:3) = E Q h e − R t L φφ ( ˆ X s ) ds φ ( ˆ X t ) φ ( ˆ X ) e R t κ ′ ( ˆ X s ) ds ( h/φ ) ′ ( ˆ X t ) (cid:12)(cid:12)(cid:12) ˆ X = x i φ ( x )= E Q h d ˆ P t d Q t e R t κ ′ ( ˆ X s ) ds ( h/φ ) ′ ( ˆ X t ) (cid:12)(cid:12)(cid:12) ˆ X = x i φ ( x )= E ˆ P (cid:2) e R t κ ′ ( ˆ X s ) ds ( h/φ ) ′ ( ˆ X t ) (cid:12)(cid:12) ˆ X = x (cid:3) φ ( x ) , hich implies that f x ( t, x ) = E ˆ P (cid:2) e R t κ ′ ( ˆ X s ) ds ( h/φ ) ′ ( ˆ X t ) (cid:12)(cid:12) ˆ X = x (cid:3) . From the time-homogeneous Markov property, f x ( T − t, x ) = E ˆ P (cid:2) e R T − t κ ′ ( ˆ X s ) ds ( h/φ ) ′ ( ˆ X T − t ) (cid:12)(cid:12) ˆ X = x (cid:3) = E ˆ P (cid:2) e R Tt κ ′ ( ˆ X s ) ds ( h/φ ) ′ ( ˆ X T ) (cid:12)(cid:12) ˆ X t = x (cid:3) so that f x ( T − t, ˆ X t ) e R t κ ′ ( ˆ X s ) ds = E ˆ P (cid:2) e R T κ ′ ( ˆ X s ) ds ( h/φ ) ′ ( ˆ X T ) (cid:12)(cid:12) F t (cid:3) . In conclusion, the process ( f x ( T − t, ˆ X t ) e R t κ ′ ( ˆ X s ) ds ) ≤ t ≤ T is a martingale. In this section, we develop the second-order initial-value sensitivity for large-time T. For agiven quadruple of functions ( b, σ, r, h ) satisfying Assumptions 2.1–2.5, the large-time asymptoticbehavior of the partial derivative ∂ ξξ p T = ∂ ξξ E P ξ [ e − R T r ( X s ) ds h ( X T )]is of interest to us.Toward this end, we need to study two quadruples in addition to the original quadruple( b, σ, r, h ) . Recall that κ = b + σ φ ′ /φ, and suppose that ( κ + σ ′ σ, σ, − κ ′ , ( h/φ ) ′ ) satisfies As-sumptions 2.1–2.5 on the consistent probability space (Ω , F , ( F t ) t ≥ , (ˆ P t ) t ≥ ) having Brownianmotion ( ˆ B t ) t ≥ as stated in Theorem 3.1. Then, we can construct the corresponding objectsˆ X, ˆ P , (ˆ λ, ˆ φ ) , ˆ M , ˆ f , (˜ P t ) t ≥ , ( ˜ B t ) t ≥ . Define γ := κ + σ ′ σ + σ ˆ φ ′ / ˆ φ . If ( γ + σ ′ σ, σ, − γ ′ , (( h/φ ) ′ / ˆ φ ) ′ ) also satisfies Assumptions 2.1 – 2.5 on the consistent prob-ability space (Ω , F , ( F t ) t ≥ , (˜ P t ) t ≥ ) having Brownian motion ( ˜ B t ) t ≥ , we can construct thecorresponding objects ˜ X, ˜ P , (˜ λ, ˜ φ ) , ˜ M , ˜ f , ( P t ) t ≥ , ( B t ) t ≥ . We summarize three quadruples and their corresponding objects in the following table.Quadruples ( b, σ, r, h ) ( κ + σ ′ σ, σ, − κ ′ , ( h/φ ) ′ ) ( γ + σ ′ σ, σ, − γ ′ , (( h/φ ) ′ / ˆ φ ) ′ )Underlying space (Ω , F , ( F t ) t ≥ , ( P t ) t ≥ ) (Ω , F , ( F t ) t ≥ , (ˆ P t ) t ≥ ) (Ω , F , ( F t ) t ≥ , (˜ P t ) t ≥ )Induced objects X, P , ( λ, φ ) , M, f, (ˆ P t ) t ≥ ˆ X, ˆ P , (ˆ λ, ˆ φ ) , ˆ M , ˆ f , (˜ P t ) t ≥ ˜ X, ˜ P , (˜ λ, ˜ φ ) , ˜ M , ˜ f , ( P t ) t ≥ Auxiliary functions κ := b + σ φ ′ /φ γ := κ + σ ′ σ + σ ˆ φ ′ / ˆ φ Sections used 3.1.1, 3.1.2, 3.2.1, 3.2.2 3.1.1, 3.1.2, 3.2.1, 3.2.2 3.1.2, 3.2.2
Table 1: Three quadruples
Before stating the rigorous results, we provide heuristic arguments to understand the moti-vation for Eq.(3.13) in Theorem 3.2. Suppose that three quadruples of functions ( b, σ, r, h ) , ( κ + σ ′ σ, σ, − κ ′ , ( h/φ ) ′ ) and ( γ + σ ′ σ, σ, − γ ′ , (( h/φ ) ′ / ˆ φ ) ′ ) satisfy Assumptions 2.1–2.5. Us-ing the Hansen–Scheinkman decomposition, we have p T = e − λT φ ( ξ ) f ( T, ξ ) and f x ( t, x ) =ˆ f ( t, x ) e − ˆ λt ˆ φ ( x ) in Eq.(3.7). Direct calculation gives ∂ ξξ p T p T − (cid:16) ∂ ξ p T p T (cid:17) − φ ′′ ( ξ ) φ ( ξ ) + (cid:16) φ ′ ( ξ ) φ ( ξ ) (cid:17) − ˆ φ ′ ( ξ )ˆ φ ( ξ ) ∂ ξ p T p T + ˆ φ ′ ( ξ )ˆ φ ( ξ ) φ ′ ( ξ ) φ ( ξ ) = ˆ f x ( T, ξ ) ˆ φ ( ξ ) e − ˆ λT f ( T, ξ ) − (cid:16) f x ( T, ξ ) f ( T, ξ ) (cid:17) . (3.11) e shift our attention to the two terms on the right-hand side. For the first term on theright-hand side, we derive the equalityˆ f x ( t, x ) = ˜ f ( t, x ) e − ˜ λt ˜ φ ( x ) , which is similar to Eq.(3.7). Thus,ˆ f x ( T, ξ ) ˆ φ ( ξ ) e − ˆ λT f ( T, ξ ) = ˜ f ( T, ξ ) e − (˜ λ +ˆ λ ) T ˜ φ ( ξ ) ˆ φ ( ξ ) f ( T, ξ ) ≃ e − (˜ λ +ˆ λ ) T . Here, for two nonzero functions h ( T ) and h ( T ), the notation h ( T ) ≃ h ( T ) implies that thelimit lim T →∞ h ( T ) h ( T ) converges to a nonzero constant. For the second term on the right-hand sideof Eq.(3.11), observe that f x ( T, ξ ) ≃ e − ˆ λT from Eq.(3.7). Thus, (cid:16) f x ( T, ξ ) f ( T, ξ ) (cid:17) ≃ e − λT . In conclusion, Eq.(3.11) satisfies ∂ ξξ p T p T − (cid:16) ∂ ξ p T p T (cid:17) − φ ′′ ( ξ ) φ ( ξ ) + (cid:16) φ ′ ( ξ ) φ ( ξ ) (cid:17) − ˆ φ ′ ( ξ )ˆ φ ( ξ ) ∂ ξ p T p T + ˆ φ ′ ( ξ )ˆ φ ( ξ ) φ ′ ( ξ ) φ ( ξ ) ≃ ( e − ˜ λT + e − ˆ λT ) e − ˆ λT , which is the motivation for Eq.(3.13).A rigorous estimation of the second-order partial derivative ∂ ξξ p T for large T is obtained byanalyzing the three quadruples stated above, as described in the following theorem. Theorem 3.2.
Suppose that σ is twice continuously differentiable and the following conditionshold.(i) The quadruple of functions ( b, σ, r, h ) satisfies Assumptions 2.1–2.5 on the consistent prob-ability space (Ω , F , ( F t ) t ≥ , ( P t ) t ≥ ) having Brownian motion ( B t ) t ≥ . (ii) The quadruple of functions ( κ + σ ′ σ, σ, − κ ′ , ( h/φ ) ′ ) satisfies Assumptions 2.1–2.5 on theconsistent probability space (Ω , F , ( F t ) t ≥ , (ˆ P t ) t ≥ ) having Brownian motion ( ˆ B t ) t ≥ . (iii) The quadruple of functions ( γ + σ ′ σ, σ, − γ ′ , (( h/φ ) ′ / ˆ φ ) ′ ) satisfies Assumptions 2.1–2.5 onthe consistent probability space (Ω , F , ( F t ) t ≥ , (˜ P t ) t ≥ ) having Brownian motion ( ˜ B t ) t ≥ . Then, for each
T > , two processes ( f x ( T − t, ˆ X t ) e R t κ ′ ( ˆ X s ) ds ) ≤ t ≤ T and ( ˆ f x ( T − t, ˜ X t ) e R t γ ′ ( ˜ X s ) ds ) ≤ t ≤ T are local martingales under the probability measures ˆ P T and ˜ P T , respectively. For each T > ,if these are martingales or if two functions f x and ˆ f x satisfy f x ( T, x ) = E ˆ P [ e R T κ ′ ( ˆ X s ) ds ( h/φ ) ′ ( ˆ X T ) | ˆ X = x ] , ˆ f x ( T, x ) = E ˜ P [ e R T γ ′ ( ˜ X s ) ds (( h/φ ) ′ / ˆ φ ) ′ ( ˜ X T ) | ˜ X = x ] , (3.12) then (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ ξξ p T p T − (cid:16) ∂ ξ p T p T (cid:17) − φ ′′ ( ξ ) φ ( ξ ) + (cid:16) φ ′ ( ξ ) φ ( ξ ) (cid:17) − ˆ φ ′ ( ξ )ˆ φ ( ξ ) ∂ ξ p T p T + ˆ φ ′ ( ξ )ˆ φ ( ξ ) φ ′ ( ξ ) φ ( ξ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ c ( e − ˜ λT + e − ˆ λT ) e − ˆ λT (3.13) for some positive constant c, which is independent of T. In particular, we have (cid:12)(cid:12)(cid:12)(cid:12) ∂ ξξ p T p T − φ ′′ ( ξ ) φ ( ξ ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ c ′ e − ˆ λT for some positive constant c ′ , which is independent of T. roof. Using the same argument as that presented in the proof of Theorem 3.1, it can be shownthat two processes ( f x ( T − t, ˆ X t ) e R t κ ′ ( ˆ X s ) ds ) ≤ t ≤ T and ( ˆ f x ( T − t, ˜ X t ) e R t γ ′ ( ˜ X s ) ds ) ≤ t ≤ T are localmartingales under the probability measures ˆ P T and ˜ P T , respectively. Thus, we omit the proof.Now, assume that two functions f x and ˆ f x satisfy Eq.(3.12). If the two local martingalesabove are martingales, then Eq.(3.12) holds by the same argument as that in Eq.(3.6). Since( κ + σ ′ σ, σ, − κ ′ , ( h/φ ) ′ ) satisfies Assumptions 2.1–2.5 on (Ω , F , ( F t ) , (ˆ P t ) t ≥ ) having Brownianmotion ( ˆ B t ) t ≥ , we can construct the corresponding objectsˆ X, ˆ P , (ˆ λ, ˆ φ ) , ˆ M , ˆ f , (˜ P t ) t ≥ , ( ˜ B t ) t ≥ and these notations are self-explanatory. The remainder functionˆ f ( t, x ) = E ˜ P x [(( h/φ ) ′ / ˆ φ )( ˆ X t )] (3.14)satisfies f x ( t, x ) = ˆ f ( t, x ) e − ˆ λt ˆ φ ( x ) by Eq.(3.7), and the dynamics of ˆ X is d ˆ X t = ( κ + σ ′ σ )( ˆ X t ) dt + σ ( ˆ X t ) d ˆ B t = γ ( ˆ X t ) dt + σ ( ˆ X t ) d ˜ B t for γ = κ + σ ′ σ + σ ˆ φ ′ / ˆ φ. Applying the Feynman–Kac formula to Eq.(3.14), we have − ˆ f t + 12 σ ( x ) ˆ f xx + γ ( x ) ˆ f x = 0 , ˆ f (0 , x ) = (( h/φ ) ′ / ˆ φ )( x ) . The function ˆ f is C , by Assumption 2.5 and every coefficient is continuously differentiable in x in the above PDE; thus, the function ˆ f is thrice differentiable in x. Taking the differentiationin x, we get − ˆ f xt + 12 σ ( x ) ˆ f xxx + ( γ + σ ′ σ )( x ) ˆ f xx + γ ′ ( x ) ˆ f x = 0 , ˆ f x (0 , x ) = (( h/φ ) ′ / ˆ φ ) ′ ( x ) . Since ( γ + σ ′ σ, σ, − γ ′ , (( h/φ ) ′ / ˆ φ ) ′ ) satisfies Assumptions 2.1–2.5, we can construct the cor-responding objects ˜ X, ˜ P , (˜ λ, ˜ φ ) , ˜ M , ˜ f , ( P t ) t ≥ , ( B t ) t ≥ . Note that the process ˜ X satisfies d ˜ X t = ( γ + σ ′ σ )( ˜ X t ) dt + σ ( ˜ X t ) d ˜ B t . The process ( ˆ f x ( T − t, ˜ X t ) e R t γ ′ ( ˜ X s ) ds ) ≤ t ≤ T is assumed to be a martingale; thus,ˆ f x ( t, x ) = E ˜ P x [ e R t γ ′ ( ˜ X s ) ds (( h/φ ) ′ / ˆ φ ) ′ ( ˜ X t )] = ˜ P T (( h/φ ) ′ / ˆ φ ) ′ ( x ) . We apply the Hansen–Scheinkman decomposition here. Since (˜ λ, ˜ φ ) is the recurrent eigenpairand ˜ f ( t, x ) = E ˜ P x [ ˜ M T ((( h/φ ) ′ / ˆ φ ) ′ / ˜ φ )( ˜ X t )] = E P x [((( h/φ ) ′ / ˆ φ ) ′ / ˜ φ )( ˜ X t )]is the remainder function, it follows, by Eq.(2.3), thatˆ f x ( t, x ) = ˜ f ( t, x ) e − ˜ λt ˜ φ ( x ) . Meanwhile, direct calculation gives ∂ ξξ p T p T − (cid:16) ∂ ξ p T p T (cid:17) − φ ′′ ( ξ ) φ ( ξ ) + (cid:16) φ ′ ( ξ ) φ ( ξ ) (cid:17) = f xx ( T, ξ ) f ( T, ξ ) − (cid:16) f x ( T, ξ ) f ( T, ξ ) (cid:17) . (3.15) e estimate the two terms on the right-hand side. Eq.(3.8) states that (cid:16) f x ( T, ξ ) f ( T, ξ ) (cid:17) ≤ c e − λT (3.16)for some positive constant c . To estimate the term f xx ( T,ξ ) f ( T,ξ ) on the right-hand side of Eq.(3.15),observe that Eq.(3.7) gives f xx ( t, x ) = ˆ f x ( t, x ) e − ˆ λt ˆ φ ( x ) + ˆ f ( t, x ) e − ˆ λt ˆ φ ′ ( x ) = ˆ f x ( t, x ) e − ˆ λt ˆ φ ( x ) + ˆ φ ′ ( x )ˆ φ ( x ) f x ( t, x ) . Combined with f x ( T,ξ ) f ( T,ξ ) = ∂ ξ p T p T − φ ′ ( ξ ) φ ( ξ ) , Eq.(3.15) becomes ∂ ξξ p T p T − (cid:16) ∂ ξ p T p T (cid:17) − φ ′′ ( ξ ) φ ( ξ ) + (cid:16) φ ′ ( ξ ) φ ( ξ ) (cid:17) − ˆ φ ′ ( ξ )ˆ φ ( ξ ) ∂ ξ p T p T + ˆ φ ′ ( ξ )ˆ φ ( ξ ) φ ′ ( ξ ) φ ( ξ )= ˆ f x ( T, ξ ) ˆ φ ( ξ ) e − ˆ λT f ( T, ξ ) − (cid:16) f ξ ( T, ξ ) f ( T, ξ ) (cid:17) = ˜ f ( T, ξ ) e − (˜ λ +ˆ λ ) T ˜ φ ( ξ ) ˆ φ ( ξ ) f ( T, ξ ) − (cid:16) f ξ ( T, ξ ) f ( T, ξ ) (cid:17) . Since f ( T, ξ ) and ˜ f ( T, ξ ) converge to nonzero constants as T → ∞ , combined with Eq.(3.16),we conclude that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ ξξ p T p T − (cid:16) ∂ ξ p T p T (cid:17) − φ ′′ ( ξ ) φ ( ξ ) + (cid:16) φ ′ ( ξ ) φ ( ξ ) (cid:17) − ˆ φ ′ ( ξ )ˆ φ ( ξ ) ∂ ξ p T p T + ˆ φ ′ ( ξ )ˆ φ ( ξ ) φ ′ ( ξ ) φ ( ξ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ c ( e − ˜ λT + e − ˆ λT ) e − ˆ λT for some positive constant c, which is independent of T. In particular, by using Eq.(3.8), wehave (cid:12)(cid:12)(cid:12)(cid:12) ∂ ξξ p T p T − φ ′′ ( ξ ) φ ( ξ ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ c ′ e − ˆ λT for some positive constant c ′ , which is independent of T. We conduct long-term sensitivity analysis with respect to perturbations of the drift and diffusionterms. The variable ǫ below is a perturbation parameter. Assumption 3.1.
Let I be an open neighborhood of and let b ( ǫ ) ( x ) , σ ( ǫ ) ( x ) , r ( ǫ ) ( x ) , h ( ǫ ) ( x ) befunctions of two variables ( ǫ, x ) on I ×D such that for each x they are continuously differentiablein ǫ on I and b (0) = b, σ (0) = σ, r (0) = r, h (0) = h. Assumption 3.2.
For each ǫ ∈ I, the quadruple ( b ( ǫ ) , σ ( ǫ ) , r ( ǫ ) , h ( ǫ ) ) satisfies Assumptions 2.1–2.5. From the above assumptions, the notations X ( ǫ ) , P ( ǫ ) , ( λ ( ǫ ) , φ ( ǫ ) ) , M ( ǫ ) , f ( ǫ ) , (ˆ P ( ǫ ) ) t ≥ , ( ˆ B ( ǫ ) t ) t ≥ are self-explanatory. The process X ( ǫ ) satisfies dX ( ǫ ) t = b ( ǫ ) ( X ( ǫ ) t ) dt + σ ( X ( ǫ ) t ) dB t = κ ( ǫ ) ( X ( ǫ ) t ) dt + σ ( X ( ǫ ) t ) d ˆ B ( ǫ ) t here κ ( ǫ ) := b ( ǫ ) + σ ( ǫ )2 φ ( ǫ ) x /φ ( ǫ ) . The perturbed pricing operator is P ( ǫ ) T h ( x ) = E P x [ e − R T r ( ǫ ) ( X ( ǫ ) s ) ds h ( X ( ǫ ) T )]and the remainder function is f ( ǫ ) ( t, x ) = E ˆ P ( ǫ ) x [( h ( ǫ ) /φ ( ǫ ) )( X ( ǫ ) t )] . (3.17) Assumption 3.3.
For each t ∈ [0 , ∞ ) and x ∈ D , the functions φ ( ǫ ) ( x ) , φ ( ǫ ) x ( x ) and the func-tions f ( ǫ ) ( t, x ) , f ( ǫ ) x ( t, x ) , f ( ǫ ) xx ( t, x ) , f ( ǫ ) t ( t, x ) are continuously differentiable in ǫ on I. For notational simplicity, we define ℓ ( ǫ ) = ∂ ǫ κ ( ǫ ) , ℓ = ℓ (0) , Σ ( ǫ ) = ∂ ǫ σ ( ǫ ) , Σ = Σ (0) . Assumption 3.4.
Three expectations E ˆ P ξ [ R t ( σ Σ)( X s ) f xx ( t − s, X s ) ds ] , E ˆ P ξ [ R t ℓ ( X s ) f x ( t − s, X s ) ds ] and E ˆ P ξ [ ∂ ǫ | ǫ =0 ( h ( ǫ ) /φ ( ǫ ) )( X t )] are uniformly bounded in t on [0 , ∞ ) . For ǫ ∈ I, consider the expectation p ( ǫ ) T = E P ξ [ e − R T r ( ǫ ) ( X ( ǫ ) s ) ds h ( ǫ ) ( X ( ǫ ) T )] = P ( ǫ ) T h ( ǫ ) ( ξ ) . We are interested in the large-time behavior of ∂∂ǫ (cid:12)(cid:12)(cid:12) ǫ =0 p ( ǫ ) T , which measures the sensitivity with respect to the perturbation parameter ǫ. The main idea isthe Hansen–Scheinkman decomposition, which allows the expectation p ( ǫ ) T to be expressed as p ( ǫ ) T = φ ( ǫ ) ( ξ ) e − λ ( ǫ ) T f ( ǫ ) ( T, ξ ) . By taking the differentiation in ǫ, after some manipulations, we have1 T ∂∂ǫ (cid:12)(cid:12)(cid:12) ǫ =0 ln p ( ǫ ) T + ∂∂ǫ (cid:12)(cid:12)(cid:12) ǫ =0 λ ( ǫ ) = 1 T ∂∂ǫ (cid:12)(cid:12) ǫ =0 φ ( ǫ ) ( ξ ) φ ( ξ ) + ∂∂ǫ (cid:12)(cid:12) ǫ =0 f ( ǫ ) ( T, ξ ) f ( T, ξ ) ! . (3.18)We will find sufficient conditions for the term ∂∂ǫ (cid:12)(cid:12) ǫ =0 f ( ǫ ) ( T, ξ ) f ( ǫ ) ( T, ξ )to be uniformly bounded in T. Then, Eq.(3.18) gives (cid:12)(cid:12)(cid:12)(cid:12) T ∂∂ǫ (cid:12)(cid:12)(cid:12) ǫ =0 ln p ( ǫ ) T + ∂∂ǫ (cid:12)(cid:12)(cid:12) ǫ =0 λ ( ǫ ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ cT (3.19)for some positive constant c, which is independent of T. Thus1
T ∂∂ǫ (cid:12)(cid:12)(cid:12) ǫ =0 ln p ( ǫ ) T → − ∂∂ǫ (cid:12)(cid:12)(cid:12) ǫ =0 λ ( ǫ ) as T → ∞ and its convergence rate is O (1 /T ) . Sufficient conditions with respect to the driftperturbation and the diffusion perturbation are investigated in Section 3.2.1 and 3.2.2, respec-tively. .2.1 Drift-term sensitivity This section investigates the long-term sensitivity with respect to a perturbation of the driftterm. We provide a sufficient condition for Eq.(3.19) to hold. Let ( b ( ǫ ) , σ, r ( ǫ ) , f ( ǫ ) ) be aquadruple of functions satisfying Assumptions 2.1 – 2.5 on the consistent probability space(Ω , F , ( F t ) t ≥ , ( P t ) t ≥ ) having Brownian motion ( B t ) t ≥ . The diffusion function σ is not per-turbed. Recall that if ( κ + σ ′ σ, σ, − κ ′ , ( h/φ ) ′ ) satisfies Assumptions 2.1 – 2.5 on the consistentprobability space (Ω , F , ( F t ) t ≥ , (ˆ P t ) t ≥ ) having Brownian motion ( ˆ B t ) t ≥ , we can construct thecorresponding objects ˆ X, ˆ P , (ˆ λ, ˆ φ ) , ˆ M , ˆ f , (˜ P t ) t ≥ , ( ˜ B t ) t ≥ . Theorem 3.3.
Suppose that the following conditions hold.(i) The quadruple ( b ( ǫ ) , σ, r ( ǫ ) , h ( ǫ ) ) satisfies Assumptions 3.1– 3.4 on the consistent probabilityspace (Ω , F , ( F t ) t ≥ , ( P t ) t ≥ ) having Brownian motion ( B t ) t ≥ . (ii) The quadruple ( κ + σ ′ σ, σ, − κ ′ , ( h/φ ) ′ ) satisfies Assumptions 2.1 – 2.5 on the consistentprobability space (Ω , F , ( F t ) t ≥ , (ˆ P t ) t ≥ ) having Brownian motion ( ˆ B t ) t ≥ . Then, for each
T > the process (cid:18) f (0) ǫ ( T − t, X t ) + Z t ℓ ( X s ) f x ( T − s, X s ) ds (cid:19) ≤ t ≤ T is a local martingale under the probability measure ˆ P T . For each
T > if this process is amartingale or if f (0) ǫ satisfies f (0) ǫ ( T, ξ ) = E ˆ P ξ h Z T ℓ ( X t ) f x ( T − t, X t ) dt + ∂ ǫ | ǫ =0 ( h ( ǫ ) /φ ( ǫ ) )( X T ) i , (3.20) then (cid:12)(cid:12)(cid:12)(cid:12) T ∂∂ǫ (cid:12)(cid:12)(cid:12) ǫ =0 ln p ( ǫ ) T + ∂∂ǫ (cid:12)(cid:12)(cid:12) ǫ =0 λ ( ǫ ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ cT for some positive constant c, which is independent of T. Proof.
Applying the Feynman–Kac formula to Eq.(3.17), we get − f ( ǫ ) t + 12 σ ( x ) f ( ǫ ) xx + κ ( ǫ ) ( x ) f ( ǫ ) x = 0 , f ( ǫ ) (0 , x ) = ( h ( ǫ ) /φ ( ǫ ) )( x )for κ ( ǫ ) = b ( ǫ ) + σ φ ( ǫ ) x /φ ( ǫ ) . Let us differentiate this PDE in ǫ and evaluate it at ǫ = 0 (Assump-tions 3.1 and 3.3). Then − f (0) ǫt + 12 σ ( x ) f (0) ǫxx + κ ( x ) f (0) ǫx + ℓ ( x ) f (0) x = 0 , f (0) ǫ (0 , x ) = ∂ ǫ | ǫ =0 ( h ( ǫ ) /φ ( ǫ ) )( x ) . Applying the Ito formula to the process ( f (0) ǫ ( T − t, X t ) + R t ℓ ( X s ) f x ( T − s, X s ) ds ) ≤ t ≤ T , itfollows that d (cid:16) f (0) ǫ ( T − t, X t ) + Z t ℓ ( X s ) f x ( T − s, X s ) ds (cid:17) = (cid:0) − f (0) ǫt + 12 σ ( X t ) f (0) ǫxx + κ ( X t ) f (0) ǫx + ℓ ( X t ) f x (cid:1) dt + σ ( X t ) f (0) ǫx d ˆ B t = σ ( X t ) f (0) ǫx d ˆ B t . (3.21)Thus, the process ( f (0) ǫ ( T − t, X t ) + R t ℓ ( X s ) f x ( T − s, X s ) ds ) ≤ t ≤ T is a local martingale underthe probability measure ˆ P T . ow, assume that f (0) ǫ ( T, ξ ) = E ˆ P ξ h Z T ℓ ( X t ) f x ( T − t, X t ) dt + ∂ ǫ | ǫ =0 ( h ( ǫ ) /φ ( ǫ ) )( X T ) i . This equality also holds if the process ( f (0) ǫ ( T − t, X t ) + R t ℓ ( X s ) f x ( T − s, X s ) ds ) ≤ t ≤ T is amartingale because f (0) ǫ ( T, ξ ) = E ˆ P ξ h Z T ℓ ( X t ) f x ( T − t, X t ) dt + f (0) ǫ (0 , X T ) i = E ˆ P ξ h Z T ℓ ( X t ) f x ( T − t, X t ) dt + ∂ ǫ | ǫ =0 ( h ( ǫ ) /φ ( ǫ ) )( X T ) i . Since two expectations E ˆ P ξ [ R T ℓ ( X t ) f x ( T − t, X t ) dt ] and E ˆ P ξ [ ∂ ǫ | ǫ =0 ( h ( ǫ ) /φ ( ǫ ) )( X T )] are uniformlybounded in T on [0 , ∞ ) by Assumption 3.4, the function f (0) ǫ ( T, ξ ) is also uniformly bounded in T on [0 , ∞ ) . Using Eq.(3.18), we obtain the desired result.
Remark 3.4.
Under the same hypothesis of Theorem 3.3, we obtain the probabilistic represen-tation f (0) ǫ ( T, ξ ) = E ˆ P ξ h Z T ℓ ( X t ) f x ( T − t, X t ) dt + ∂ ǫ | ǫ =0 ( h ( ǫ ) /φ ( ǫ ) )( X T ) i presented in Eq. (3.20) if E ˆ P ξ h Z T ( σ ( X t ) f (0) ǫx ( T − t, X t )) dt i < ∞ . This is evident from Eq. (3.21) because the process ( f (0) ǫ ( T − t, X t )+ R t ℓ ( X s ) f x ( T − s, X s ) ds ) ≤ t ≤ T is a martingale under the probability measure ˆ P T . This section investigates the long-term sensitivity with respect to a perturbation of the diffusionterm. We provide a sufficient condition for Eq.(3.19) to hold. Let ( b ( ǫ ) , σ ( ǫ ) , r ( ǫ ) , f ( ǫ ) ) be aquadruple of functions satisfying Assumptions 3.1 – 3.4 on the consistent probability space(Ω , F , ( F t ) t ≥ , ( P t ) t ≥ ) having Brownian motion ( B t ) t ≥ . The three quadruples listed in Table1 are used in this section. We recall that Σ ( ǫ ) = ∂ ǫ σ ( ǫ ) and Σ = Σ (0) . Theorem 3.4.
Suppose that the following conditions hold.(i) The quadruple ( b ( ǫ ) , σ ( ǫ ) , r ( ǫ ) , f ( ǫ ) ) satisfies Assumptions 3.1– 3.4 on the consistent proba-bility space (Ω , F , ( F t ) t ≥ , ( P t ) t ≥ ) having Brownian motion ( B t ) t ≥ . (ii) The quadruple ( κ + σ ′ σ, σ, − κ ′ , ( h/φ ) ′ ) satisfies Assumptions 2.1 – 2.5 on the consistentprobability space (Ω , F , ( F t ) t ≥ , (ˆ P t ) t ≥ ) having Brownian motion ( ˆ B t ) t ≥ . (iii) The quadruple ( γ + σ ′ σ, σ, − γ ′ , (( h/φ ) ′ / ˆ φ ) ′ ) satisfies Assumptions 2.1 – 2.5 on the consistentprobability space (Ω , F , ( F t ) t ≥ , (˜ P t ) t ≥ ) having Brownian motion ( ˜ B t ) t ≥ . Then, for each
T > the process (cid:16) f (0) ǫ ( T − t, X t ) + Z t (( σ Σ)( X s ) f xx ( T − s, X s ) + ℓ ( X s ) f x ( T − s, X s )) ds (cid:17) ≤ t ≤ T s a local martingale under the probability measure ˆ P T . For each
T > if this process is amartingale or if f (0) ǫ satisfies f (0) ǫ ( T, ξ ) = E ˆ P ξ h Z T (( σ Σ)( X t ) f xx ( T − t, X t ) + ℓ ( X t ) f x ( T − t, X t )) dt i + E ˆ P ξ [ ∂ ǫ | ǫ =0 ( h ( ǫ ) /φ ( ǫ ) )( X T )] , then (cid:12)(cid:12)(cid:12)(cid:12) T ∂∂ǫ (cid:12)(cid:12)(cid:12) ǫ =0 ln p ( ǫ ) T + ∂∂ǫ (cid:12)(cid:12)(cid:12) ǫ =0 λ ( ǫ ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ cT for some positive constant c, which is independent of T. Proof.
Applying the Feynman–Kac formula to Eq.(3.17), we get − f ( ǫ ) t + 12 σ ( ǫ )2 ( x ) f ( ǫ ) xx + κ ( ǫ ) ( x ) f ( ǫ ) x = 0 , f ( ǫ ) (0 , x ) = ( h ( ǫ ) /φ ( ǫ ) )( x )for κ ( ǫ ) = b ( ǫ ) + σ ( ǫ )2 φ ( ǫ ) x /φ ( ǫ ) . Let us differentiate this PDE in ǫ and evaluate it at ǫ = 0 , then − f (0) ǫt + 12 σ ( x ) f (0) ǫxx + κ ( x ) f (0) ǫx + ( σ Σ)( x ) f (0) xx + ℓ ( x ) f (0) x = 0 , f (0) ǫ (0 , x ) = ∂ ǫ | ǫ =0 ( h ( ǫ ) /φ ( ǫ ) )( x ) . From the Ito formula, one can show that the process (cid:16) f (0) ǫ ( T − t, X t ) + Z t (( σ Σ)( X s ) f xx ( T − s, X s ) + ℓ ( X s ) f x ( T − s, X s )) ds (cid:17) ≤ t ≤ T is a local martingale under the probability measure ˆ P T by checking that the dt term vanishes.Suppose that f ǫ satisfies f (0) ǫ ( T, ξ ) = E ˆ P ξ h Z T (( σ Σ)( X t ) f xx ( T − t, X t ) + ℓ ( X t ) f x ( T − t, X t )) dt i + E ˆ P ξ [ ∂ ǫ | ǫ =0 ( h ( ǫ ) /φ ( ǫ ) )( X T )] . This equality holds if the local martingale above is a martingale. Since three expectations E ˆ P ξ [ R T ( σ Σ)( X t ) f xx ( T − t, X t ) dt ] , E ˆ P ξ [ R T ℓ ( X t ) f x ( T − t, X t ) dt ] and E ˆ P ξ [ ∂ ǫ | ǫ =0 ( h ( ǫ ) /φ ( ǫ ) )( X T )] areuniformly bounded in T on [0 , ∞ ) by Assumption 3.4, the function f (0) ǫ ( T, ξ ) is also uniformlybounded in T on [0 , ∞ ) . Using Eq.(3.18), we obtain the desired result.
We present applications of the previous results to three practical problems: utility maximization,entropic risk measures and bond prices.
This section discusses the classical utility maximization problem in complete markets as an ap-plication. We first describe the general Ito process models to formulate the utility maximizationproblem. Then, more specific models, namely factor models and local volatility models, arediscussed in Sections 4.1.1 and 4.1.2, respectively.Let (Ω , F , ( F t ) t ≥ , ( P t ) t ≥ ) be a consistent probability space having a d -dimensional Brow-nian motion Z = ( Z (1) t , · · · , Z ( d ) t ) ⊤ t ≥ . The filtration ( F t ) t ≥ satisfies the usual condition. Theprobability measures ( P t ) t ≥ are referred to as the physical measures of the market. An Itoprocess model is describes as follows. i) The bank account, denoted as ( G t ) t ≥ , is a process given by G t = e R t r u du , t ≥ r t ) t ≥ is a progressively measurable process taking values in [0 , ∞ ) such that R T r u du < ∞ P T -almost surely for each T ≥ . (ii) There are d stocks ( S t ) t ≥ = ( S (1) t , · · · , S ( d ) t ) ⊤ t ≥ described as S ( i ) t = S ( i )0 e R t ( µ ( i ) u − | σ ( i ) u | ) du + R t σ ( i ) u dZ u , S ( i )0 > i = 1 , , · · · , d, where ( µ ( i ) t ) t ≥ is a progressively measurable process with R T | µ ( i ) u | du < ∞ P T -almost surely for each T ≥ , and ( σ ( i ) t ) t ≥ is a d -dimensional progressively mea-surable row process with R T | σ ( i ) u | du < ∞ P T -almost surely for each T ≥ . For simpler expressions, we define a d -dimensional column process( µ t ) t ≥ := ( µ (1) t , · · · , µ ( d ) t ) ⊤ t ≥ and a d × d -matrix process ( σ t ) t ≥ = σ (1) t σ (2) t ... σ ( d ) t . In the SDE form, we can write dG t = r t G t dt and dS t = D ( S t ) µ t dt + D ( S t ) σ t dZ t where D ( x ) for x = ( x , · · · , x d ) ⊤ is the diagonal matrix whose i -th diagonal entry is x i for i = 1 , · · · , d and off-diagonal entries are zero.A self-financing portfolio is a pair ( x, π ) of a real number x and a d -dimensional progressivelymeasurable process ( π t ) t ≥ that is ( S t /G t ) ≤ t ≤ T -integrable for each T ≥ . The value processΠ = (Π t ) t ≥ = (Π ( x,π ) t ) t ≥ of the portfolio ( x, π ) is given byΠ ( x,π ) t = xG t + G t Z t π u d ( S/G ) u . (4.3)For a positive number x, let X T ( x ) denote the family of nonnegative value processes with initialvalue x , i.e., X T ( x ) = (cid:8) (Π ( x,π ) t ) ≤ t ≤ T : ∃ a self-financing portfolio ( x, π ) such that Π ( x,π ) t ≥ ≤ t ≤ T (cid:9) . Define X T = X T (1) . (4.4)We now construct a consistent family of risk-neutral measures ( Q t ) t ≥ . Assumption 4.1.
Consider the following conditions.(i) The d × d -matrix process σ : [0 , T ] × Ω → R d × d is invertible Leb ⊗ P T -almost surely foreach T ≥ . ii) The process θ t := σ − t ( µ t − r t ) , ≤ t ≤ T (4.5) is square-integrable, i.e., R T | θ u | du < ∞ , P T -almost surely for each T ≥ . This process θ is referred to as the market price of risk.(iii) A local martingale ( e − R t θ u dZ u − R t | θ u | du ) ≤ t ≤ T is a martingale under the probability measure P T for each T ≥ . Remark 4.1.
The fundamental theorem of asset pricing states that Assumption 4.1 is equivalentto the NFLVR (no free lunch with vanishing risk) condition on time interval [0 , T ] for each T > .Refer to Delbaen and Schachermayer (1994). Define a probability measure Q T on F T by L T := d Q T d P T = e − R T θ u dZ u − R T | θ u | du . (4.6)Then, the probability measure Q T is equivalent to P T and the processes S (1) /G, · · · , S ( d ) /G are Q T -local martingales. As is well known, this measure Q T is called the risk-neutral measure . Itis evident that the family of risk-neutral measures ( Q t ) t ≥ is consistent. Define a process W as W t = Z t θ u du + Z t , t ≥ . Then, W is a Brownian motion on the consistent probability space (Ω , F , ( F t ) t ≥ , ( Q t ) t ≥ ) bythe Girsanov theorem.In this market, an agent has a utility function U : [0 , ∞ ) → R for wealth. Assume that theutility function is a power function of the form u ( x ) = x ν /ν for ν < . This utility function is increasing, strictly concave, continuously differentiable andsatisfies the Inada conditions. For given unit initial endowment, the agent wants to maximizethe expected utility max Π ∈X T E P [ U (Π T )] , of which the long-term sensitivity with respect to small changes of the underlying process is ofinterest to us.Kramkov and Schachermayer (1999) states that the optimal portfolio value isΠ (opt) T = c T ( U ′ ) − ( L T /G T ) = c T ( L T /G T ) / ( ν − where c T is a constant satisfying the budget constraint1 = E Q [Π (opt) T /G T ] = c T E Q [( L T /G νT ) / ( ν − ] . Thus, the optimal expected utility is E P [ U (Π (opt) T )] = E P [ U ( c T ( L T /G T ) / ( ν − )] = 1 ν c νT E P [( L T /G T ) ν/ ( ν − ]= 1 ν E P [( L T /G T ) ν/ ( ν − ]( E Q [( L T /G νT ) / ( ν − ]) ν = 1 ν E Q [( L T /G νT ) / ( ν − ]( E Q [( L T /G νT ) / ( ν − ]) ν = 1 ν ( E Q [( L T /G νT ) / ( ν − ]) − ν . efining u T = E Q [( L T /G νT ) / ( ν − ] , the optimal expected utility satisfiesmax Π ∈X E P [ U (Π T )] = E P [ U (Π (opt) T )] = u − νT /ν . Thus, the problem of studying the long-term sensitivity of the optimal expected utility boilsdown to analyzing the expectation u T . We can analyze the large-time behavior of u T for small changes of the underlying process asfollows. From Eq.(4.6), the Radon–Nikodym derivative L T is L T = e − R T θ u dZ u − R T | θ u | du = e − R T θ u dW u + R T | θ u | du , thus u T = E Q [( L T /G νT ) / ( ν − ] = E Q [ e − ν − R T θ u dW u + ν − R T | θ u | du − νν − R T r u du ]= E P [ e ν ν − R T | θ u | du − νν − R T r u du ]where ( P t ) t ≥ is a consistent family of probability measures defined from ( Q t ) t ≥ by the Girsanovkernel ν − θ, i.e., P t is a probability measure on F t given as d P t d Q t = e − ν − R t θ u dW u − ν − R t | θ u | du . (4.7)Here, the following condition was assumed for this change of measures. Assumption 4.2.
A local martingale ( e − ν − R t θ u dW u − ν − R t | θ u | du ) ≤ t ≤ T is a martingale under the probability measure Q T for each T ≥ . Then, a process B defined by B t := 1 ν − Z t θ u du + W t , t ≥ , F , ( F t ) t ≥ , ( P t ) t ≥ ) . We can summarize the arguments of this section as follows.
Proposition 4.1.
Consider the Ito process model with the bank account and the d stocks statedin Eq. (4.1) and Eq. (4.2) , respectively. Under Assumptions 4.1 and 4.2, we define a consistentfamily of probability measures ( P t ) t ≥ by Eq. (4.7) . Then, for the power utility function u ( x ) = x ν /ν, ν < and the family of nonnegative wealth processes X in Eq. (4.4) , the optimal expectedutility is max Π ∈X E P [ U (Π T )] = u − νT /ν where u T = E P [ e ν ν − R T | θ u | du − νν − R T r u du ] . (4.9) .1.1 Factor models In this section, we investigate the long-term sensitivity of the optimal expected utility un-der one-factor models driven by a multi-dimensional Brownian motion. Recall that Z =( Z (1) t , · · · , Z ( d ) t ) ⊤ t ≥ is a d -dimensional Brownian motion on a consistent probability space (Ω , F , ( F t ) t ≥ , ( P t ) t ≥ ) . The probability measures ( P t ) t ≥ are referred to as physical measures of the market.(i) Let D be an open interval in R . A factor process is a one-dimensional Markov diffusionprocess X with state space D satisfying dX t = k ( X t ) dt + v ( X t ) dZ t , X = ξ (4.10)where the function k : D → R and the row vector function v : D → R d are continuouslydifferentiable.(ii) The process ( r t ) t ≥ in Eq.(4.1) is given as r t = r ( X t ) (4.11)for continuous function r ( · ) : D → R . (iii) The processes ( µ t ) t ≥ and ( σ t ) t ≥ in Eq.(4.2) are given as µ t = µ ( X t ) , σ t = σ ( X t ) (4.12)for continuously differentiable functions µ ( · ) : D → R d and σ ( · ) : D → R d × d . This gives the full description of the one-factor model.The long-term sensitivity of the optimal expected utility in a factor model can be manipulatedto fit the underlying framework of this paper. As discussed in Proposition 4.1, it suffices toanalyze the expectation u T in Eq.(4.9). Under Assumptions 4.1 and 4.2, we have u T = E P [ e ν ν − R T | θ ( X u ) | du − νν − R T r ( X u ) du ]where θ : D → R is a function defined as θ ( · ) = σ − ( · )( µ ( · ) − r ( · ) ) . Note that the process( θ ( X t )) t ≥ is the market price of risk presented in Eq.(4.5). Observe that the process ( B t ) t ≥ inEq.(4.8) is a Brownian motion under the consistent family of probability measures ( P t ) t ≥ andthat X satisfies dX t = k ( X t ) dt + v ( X t ) dZ t = ( k − vθ )( X t ) dt + v ( X t ) dW t = (cid:0) k − νν − vθ (cid:1) ( X t ) dt + v ( X t ) dB t . By defining r ( · ) = − ν ν − | θ ( · ) | + νν − r ( · ) , the expectation u T is expressed as u T = E P [ e − R T r ( X u ) du ] . Thus, the quadruple of functions (cid:16) k ( · ) − νν − v ( · ) θ ( · ) , | v ( · ) | , r ( · ) , (cid:17) (4.13)and the above expectation u T fit the underlying framework of this paper on the consistentprobability space (Ω , F , ( F t ) t ≥ , ( P t ) t ≥ ) having Brownian motion( B t ) t ≥ := (cid:16) ν − Z t θ u du + W t (cid:17) t ≥ . We now investigate several specific examples. As one can see, the utility maximizationproblem is specified by the factor process X, the short rate function r and the function θ representing the market price of risk. Thus, it is more convenient to specify these rather thanthe functions µ and σ in Eq.(4.12). xample 4.1. (The Heston model) Let D = (0 , ∞ ) . Assume that the short rate is zero and thestock price follows dS t = µX t S t dt + p X t S t dZ (1) t , S > dX t = k ( m − X t ) dt + vρ p X t dZ (1) t + v p − ρ p X t dZ (2) t , X = ξ and that the market price of risk is θ ( X t ) = ( µ √ X t , ⊤ . Here, the parameters satisfy µ, v ∈ R ,k, m, ξ > and − ≤ ρ ≤ . Then, the quadruple of functions in Eq. (4.13) is (cid:16) km − ( k + νν − vρµ ) x, v √ x, − νµ ν − x + νrν − , (cid:17) , x ∈ D and the optimal expected utility u T is u T = E P [ e νµ ν − R T X u du ] e − νrν − T . We can simplify this problem as follows. Define a = k + νν − vρµ , b = km , σ = v , q = − νµ ν − and p T := E P [ e − q R T X u du ] (thus, u T = p T e − νrν − T ). Then, the quadruple of functions ( b − ax, σ √ x, qx, , x ∈ D and the expectation p T fit the underlying framework of this paper. The details of the long-termsensitivities are discussed in Section 5.1. Example 4.2. (The 3 / Let D = (0 , ∞ ) . Assume that the short rate is zero and thestock price follows dS t = µX t S t dt + p X t S t dZ (1) t , S > dX t = k ( m − X t ) X t dt + vρX / t dZ (1) t + v p − ρ X / t dZ (2) t , X = ξ > and that the market price of risk is θ ( X t ) = ( µ √ X t , ⊤ . Then, the quadruple of functions inEq. (4.13) is (cid:16) km − ( k + νν − vρµ ) x , vx / , − νµ ν − x + νrν − , (cid:17) , x ∈ D and the optimal expected utility u T is u T = E P [ e νµ ν − R T X u du ] e − νrν − T . We can simplify this problem as follows. Define a = k + νν − vρµ , b = km , σ = v , q = − νµ ν − and p T := E P [ e − q R T X u du ] (thus, u T = p T e − νrν − T ). Then, the quadruple of functions ( b − ax , σx / , qx, , x ∈ D and the expectation p T fit the underlying framework of this paper. The details of the long-termsensitivities are discussed in Section 5.2. .1.2 Local volatility models We investigate the utility maximization problem in local volatility models. Consider a consistentprobability space (Ω , F , ( F t ) t ≥ , ( P t ) t ≥ ) having one-dimensional Brownian motion Z = ( Z t ) t ≥ . The filtration ( F t ) t ≥ satisfies the usual condition. The probability measures ( P t ) t ≥ are referredto as physical measures of the market. A local volatility model is described as follows.(i) Assume that the short interest rate in Eq.(4.1) is a constant r ≥ . (ii) Let D = (0 , ∞ ) . A stock price is a Markov diffusion process ( S t ) t ≥ with state space D . Assume that S satisfies dS t S t = µ ( S t ) dt + σ ( S t ) dZ t for continuously differentiable functions µ ( · ) : D → R and σ ( · ) : D → R . In other words,the processes ( µ t ) t ≥ and ( σ t ) t ≥ in Eq.(4.2) are given as µ t = µ ( S t ) , σ t = σ ( S t ) . (4.14)This gives the full description of the local volatility model used in this paper.The long-term sensitivity of the optimal expected utility can be analyzed by the same ar-gument in Section 4.1.1. Under Assumptions 4.1 and 4.2, the optimal expected utility u T inProposition 4.1 is u T = E P [ e ν ν − R T θ ( S u ) du ] e − νrν − T where θ ( · ) := σ − ( · )( µ ( · ) − r ) . For convenience, we define p T := E P [ e ν ν − R T θ ( S u ) du ]so that u T = p T e − νrν − T . The process( B t ) t ≥ := (cid:16) νν − Z t θ u du + Z t (cid:17) t ≥ is a Brownian motion on the consistent probability space (Ω , F , ( F t ) t ≥ , ( P t ) t ≥ ), and the stockprice S follows dS t S t = (cid:0) − ν − µ ( S t ) + νrν − (cid:1) dt + σ ( S t ) dB t . Thus, the quadruple of functions (cid:16) − ν − µ ( · ) · + νrν − , σ ( · ) · , − ν ν − θ ( · ) , (cid:17) (4.15)and the above expectation p T fit the underlying framework of this paper on the consistentprobability space (Ω , F , ( F t ) t ≥ , ( P t ) t ≥ ) having Brownian motion ( B t ) t ≥ . Example 4.3. (The CEV model)
Let D = (0 , ∞ ) . Assume that the stock price follows the CEVmodel, which is given as a solution of dS t S t = k dt + σS tβ dB t , X = ξ for β, k, σ, ξ > . Then, the market price of risk is θ ( S t ) = k − rσ S − βt . The quadruple of functionsin Eq. (4.15) is (cid:16) νr − kν − x, σx β +1 , − ( k − r ) ν σ ( ν − x − β , (cid:17) , x ∈ D nd the expectation p T is p T = E P [ e ( k − r )2 ν σ ν − R T S − βu du ] (thus, the optimal expected utility is u T = p T e − νrν − T ). We can simplify this problem as follows.Define µ = νr − kν − , q = − ( k − r ) ν σ ( ν − . Then, the quadruple of functions ( µx, σx β +1 , qx − β , , x ∈ D and the expectation p T = E P [ e − q R T S − βu du ] fit the underlying framework of this paper. Thedetails of the long-term sensitivities are discussed in Section 5.3. In this section, we investigate the long-term sensitivity of entropic risk measures. The entropicrisk measure of a portfolio value Π T is defined as ρ (Π T ) = 1 ν ln E ( e − ν Π T )for the risk aversion parameter ν > . The main purpose of this section is to measure the extentto which the entropic risk measure is affected by small perturbations of the underlying model.First, we discuss how to formulate the entropic risk measure under general Ito process models.Then, we consider specific models in Sections 4.2.1 and 4.2.2.The entropic risk measure can be expressed in a manageable manner as follows. Recall theIto process model described in Eq.(4.1) and Eq.(4.2). In this section, the short rate is assumedto be zero so that the bank account in Eq.(4.1) is identically equal to one. Without loss ofgenerality, we consider only portfolios with zero initial capital. For given self-financing portfolio π, the value process (Π t ) t ≥ = (Π ( π ) t ) t ≥ isΠ ( π ) t = Z t π u dS u (4.16)as presented in Eq.(4.3). Then, ρ (Π T ) = 1 ν ln E P ( e − ν Π T )= 1 ν ln E P ( e − ν R T π u D ( S u ) µ u du − ν R T π u D ( S u ) σ u dZ u )= 1 ν ln E P ( e − ν R T ( π u D ( S u ) µ u − ν | π u D ( S u ) σ u | ) du )where ( P t ) t ≥ is a consistent family of probability measures defined by d P t d P t = e − ν R t π u D ( S u ) σ u dZ u − ν R t | π u D ( S u ) σ u | du . (4.17)Here, the following condition was assumed for this change of measures. Assumption 4.3.
A local martingale ( e − ν R t π u D ( S u ) σ u dZ u − ν R t | π u D ( S u ) σ u | du ) ≤ t ≤ T is a martingale under the physical measure P T for each T ≥ . e can summarize the above-mentioned arguments as follows. Proposition 4.2.
Consider the Ito process model with zero short rate and the d stocks inEq. (4.2) , and let Π ( π ) T be the value at time T of a portfolio π presented in Eq. (4.16) . UnderAssumption 4.3, we define a consistent family of probability measures ( P t ) t ≥ by Eq. (4.17) .Then, the entropic risk measure of Π ( π ) T is ρ (Π ( π ) T ) = 1 ν ln E P ( e − ν R T ( π u D ( S u ) µ u − ν | π u D ( S u ) σ u | ) du ) . (4.18) The process B t := Z t + ν Z t σ ⊤ u D ( S u ) π ⊤ u du , t ≥ is a Brownian motion on the consistent probability space (Ω , F , ( F t ) t ≥ , ( P t ) t ≥ ) . We consider the factor model as a specific case to investigate the entropic risk measure ofportfolios. The factor process X is given by Eq.(4.10), dX t = k ( X t ) dt + v ( X t ) dZ t , X = ξ, and the drift and volatility functions, µ ( · ) and σ ( · ), respectively, are given by Eq.(4.12). Theshort rate in Eq.(4.11) is assumed to be zero. We consider a portfolio π such that π t D ( S t )is determined by the factor X t ; more precisely, there is a continuously differentiable function η : R → R d such that π t D ( S t ) = η ( X t ) , t ≥ η ( X ) D − ( S ) is S -integrable. By Eq.(4.18), the entropic risk measure is ρ (Π ( π ) T ) = 1 ν ln E P ( e − ν R T ( η ( X u ) µ ( X u ) − ν | η ( X u ) σ ( X u ) | ) du )= 1 ν ln E P ( e − ν R T r ( X u ) du )where r ( · ) = η ( · ) µ ( · ) − ν | η ( · ) σ ( · ) | and the factor process X satisfies dX t = ( k ( X t ) − νv ( X t ) σ ⊤ ( X t ) η ⊤ ( X t )) dt + v ( X t ) dB t for ( P t ) t ≥ -Brownian motion ( B t ) t ≥ . In conclusion, the quadruple of functions (cid:16) k ( · ) − νv ( · ) σ ⊤ ( · ) η ⊤ ( · ) , | v ( · ) | , ν r ( · ) , (cid:17) (4.19)and the expectation u T := E P [ e − ν R T r ( X u ) du ] (thus, the entropic risk measure is ρ (Π ( π ) T ) = ν ln u T ) fit the underlying framework of this paper. Example 4.4. (Constant proportion portfolios in an affine model)
We consider the entropicrisk measure of a constant proportion portfolio η ( · ) = η = ( η , · · · , η d ) ∈ R d in an affine model.Assume that the factor process X satisfies dX t = k ( m − X t ) dt + √ X t v dZ t , X = ξ or k, m, ξ > and v = ( v , · · · , v d ) ∈ R d and that the drift and the volatility of the d stocksdescribed in Eq. (4.12) are given as µ ( X t ) := ( µ i + γ i X t ) ≤ i ≤ d , σ ( X t ) := ( p δ ij + ς ij X t ) ≤ i,j ≤ d for constants µ i , γ i ∈ R , δ ij , ς ij ≥ , ≤ i, j ≤ d. Then, the quadruple in Eq. (4.19) is (cid:16) k ( m − x ) − νv √ xσ ⊤ ( x ) η ⊤ , | v |√ x, ν ( ηµ ( x ) − ν | ησ ( x ) | ) , (cid:17) , x > and the entropic risk measure is ρ (Π ( π ) T ) = ν ln u T where u T := E P ( e − ν R T ( ηµ ( X u ) − ν | ησ ( X u ) | ) du ) . As a specific example, let δ ij = 0 for ≤ i, j ≤ d. Then, this problem can be simplified as follows.Define b = mk, a = k + ν d X i,j =1 v i √ ς ij η j , σ = | v | , q = ν (cid:0) d X i =1 η i γ i − ν X j (cid:0) X i η i √ ς ij (cid:1) (cid:1) , and we consider the case q > . The quadruple of functions ( b − ax, σ √ x, qx, , x > and the expectation p T := E P [ e − q R T X u du ] (thus, u T = p T e − T ν P di =1 η i µ i ) fit the underlying frame-work of this paper. The details of the long-term sensitivities are discussed in Section 5.1. This section investigates the entropic risk measure of portfolios in the local volatility modelpresented in Eq.(4.14). We consider portfolios determined by the stock price, i.e., π t = π ( S t )for continuously differentiable function π ( · ) . By Eq.(4.18), the entropic risk measure is ρ (Π ( π ) T ) = 1 ν ln E P ( e − ν R T ( π ( S u ) µ ( S u ) S u − νπ ( S u ) σ ( S u ) S u ) du )= 1 ν ln E P ( e − ν R T r ( S u ) du )where r ( · ) = π ( · ) µ ( · ) · − νπ ( · ) σ ( · ) · and the stock price process S satisfies dS t = ( µ ( S t ) − νπ ( S t ) σ ( S t ) S t ) S t dt + σ ( S t ) S t dB t (4.20)for ( P t ) t ≥ -Brownian motion ( B t ) t ≥ . In conclusion, the quadruple of functions (cid:16) ( µ ( · ) − νπ ( · ) σ ( · ) · ) · , σ ( · ) · , r ( · ) , (cid:17) (4.21)and the expectation u T := E P [ e − ν R T r ( X u ) du ] fit the underlying framework of this paper. Example 4.5. (Constant proportion portfolios I)
We investigate the entropic risk measure ofa portfolio in the / model. Assume that the stock price follows dS t = k ( m − S t ) S t dt + vS / t dZ t , S > or m, k, ξ > and v = 0 . In this example, we consider a constant proportion portfolio π ( · ) · = η ∈ R satisfying < η < kmνv . Then, the quadruple in Eq. (4.21) is (cid:16) ( km − νv η − kx ) x, vx / , − kη + η ( km − νv η ) x − , (cid:17) , x > and the entropic risk measure is ρ (Π ( π ) T ) = ν ln u T where u T := E P ( e − νη ( km − νv η ) R T S − u du ) e νkηT . It is noteworthy that the process S satisfies dS t = ( km − νv η − kS t ) S t dt + vS / t dB t as presented in Eq. (4.20) and the process X := 1 /S satisfies dX t = ( k + v − ( km − νv η ) X t ) dt − v p X t dB t . This problem can be simplified on the basis of the process X := 1 /S being a CIR model. Define b = k + v , a = km − νv η, σ = − v, q = νη ( km − νv η ) , then the quadruple of functions ( b − ax, σ √ x, qx, , x > and the expectation p T := E P [ e − q R T X u du ] (thus, u T = p T e νkηT ) fit the underlying framework ofthis paper. The details of the long-term sensitivities are discussed in Section 5.1. Example 4.6. (Constant proportion portfolios II)
We investigate the entropic risk measure ofanother portfolio in the / model. Assume that the stock price follows Eq. (4.22) for m, k, ξ > and v = 0 . In this example, we consider a constant proportion portfolio π ( · ) · = η ∈ R satisfying − kνv < η < . Then, the quadruple in Eq. (4.21) is (cid:16) ( km − ( k + νv η ) x ) x, vx / , mkη − η ( k + 12 νv η ) x, (cid:17) , x > and the entropic risk measure is ρ (Π ( π ) T ) = ν ln u T where u T := E P ( e νη ( k + νv η ) R T S u du ) e − νmkηT . We can simplify this problem as follows. Define a = k + νv η , b = km , σ = v , q = − νη ( k + 12 νv η ) . Then, the quadruple of functions ( b − ax , σx / , qx, , x ∈ D and the expectation p T := E P [ e − q R T S u du ] (thus, u T = p T e − νmkηT ) fit the underlying frameworkof this paper. The details of the long-term sensitivities are discussed in Section 5.2. .3 Bond prices In this section, we study the long-term sensitivity of bond prices whose underlying short rate ismodeled by a Markov diffusion. Let ( P t ) t ≥ be a consistent family of risk-neutral measures andlet X be a short interest rate process given by dX t = k ( X t ) dt + v ( X t ) dB t , X = ξ for a ( P t ) t ≥ -Brownian motion ( B t ) t ≥ . Then, the bond price with maturity T is given by p T := E P [ e − R T X s ds ] . The quadruple of functions ( k ( · ) , v ( · ) , · ,
1) and p T satisfy the underlying framework of thispaper. Example 4.7. (The CIR model)
Assume that the short rate follows the CIR model dX t = ( b − aX t ) dt + σ p X t dB t , X = ξ for a, ξ > , σ = 0 , b > σ . Then, the bond price is p T = E P [ e − R T X s ds ] . The long-termsensitivity of p T is analyzed in Section 5.1. Example 4.8. (The 3 / Assume that the short rate follows the / model dX t = ( b − aX t ) X t dt + σX t / dB t , X = ξ for b, σ, ξ > and a > − σ / . Then, the bond price is p T = E P [ e − R T X s ds ] . The long-termsensitivity of p T is analyzed in Section 5.2. Concrete examples are studied in this section. We describe three specific models: the CIRmodel, the 3/2 model and the CEV model.
We consider the CIR model dX t = ( b − aX t ) dt + σ p X t dB t , X = ξ for a, ξ > , σ = 0 , b > σ . As is well known, the process X stays positive, thus we put thedomain D = (0 , ∞ ) . The expectation p T of our interest is p T = E P ξ [ e − q R T X s ds h ( X T )]where q > h : D → R is a nonzero, nonnegative, twice differentiable function withpolynomial growth h, h ′ , h ′′ . The process X and the expectation p T are specified by the quadrupleof functions ( b − ax, σ √ x, qx, h ( x )) , x ∈ D . We are interested in the large-time behavior of p T for small perturbations of the parameters ξ, b, a and σ. It can be shown that the corresponding recurrent eigenpairs are( λ, φ ( x )) = ( bη, e − ηx ) , (ˆ λ, ˆ φ ( x )) = (˜ λ, ˜ φ ( x )) = ( α, α := p a + 2 qσ , η := α − aσ . or the long-term first-order and second-order sensitivities with respect to the initial-value, wehave (cid:12)(cid:12)(cid:12)(cid:12) ∂ ξ p T p T + η (cid:12)(cid:12)(cid:12)(cid:12) ≤ ce − αT , (cid:12)(cid:12)(cid:12)(cid:12) ∂ ξξ p T p T − η (cid:12)(cid:12)(cid:12)(cid:12) ≤ ce − αT for some positive constant c. We can also provide a higher-order convergence rate of the second-order sensitivity as (cid:12)(cid:12)(cid:12)(cid:12) ∂ ξξ p T p T − (cid:16) ∂ ξ p T p T (cid:17) (cid:12)(cid:12)(cid:12)(cid:12) ≤ ce − αT for some positive constant c. The long-term sensitivities with respect to the parameters b, a and σ are described as (cid:12)(cid:12)(cid:12)(cid:12) T ∂ b p T p T + η (cid:12)(cid:12)(cid:12)(cid:12) ≤ cT (cid:12)(cid:12)(cid:12)(cid:12) T ∂ a p T p T + bσ (cid:16) aα − (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) ≤ cT (cid:12)(cid:12)(cid:12)(cid:12) T ∂ σ p T p T + 2 b (cid:16) qασ − α − aσ (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) ≤ cT for some positive constant c. For the proofs of these asymptotic behaviors, see Appendix A.
In this section, we consider the 3 / dX t = ( b − aX t ) X t dt + σX t / dB t , X = ξ for b, σ, ξ > a > − σ / . As is well known, the process X stays positive, thus we put thedomain D = (0 , ∞ ) . The expectation p T of our interest is p T = E P ξ [ e − q R T X s ds h ( X T )]where q > h : D → R is a nonzero, nonnegative, twice differentiable function withpolynomial growth h, h ′ , h ′′ . The process X and the expectation p T are specified by the quadrupleof functions (( b − ax ) x, σx / , qx, h ( x )) , x ∈ D . We are interested in the large-time behavior of p T for small perturbations of the parameters ξ, b, a and σ. It can be shown that the corresponding recurrent eigenpairs are( λ, φ ( x )) := ( bη, x − η ) , (ˆ λ, ˆ φ ( x )) = (˜ λ, ˜ φ ( x )) = ( b, x − )where η := p ( a + σ / + 2 qσ − ( a + σ / σ . For the long-term first-order and second-order sensitivities with respect to the initial-value, wehave (cid:12)(cid:12)(cid:12)(cid:12) ∂ ξ p T p T + ηξ (cid:12)(cid:12)(cid:12)(cid:12) ≤ ce − bT , (cid:12)(cid:12)(cid:12)(cid:12) ∂ ξξ p T p T − η ( η + 1) ξ (cid:12)(cid:12)(cid:12)(cid:12) ≤ ce − bT for some positive constant c. We can also provide a higher-order convergence rate of the second-order sensitivity as (cid:12)(cid:12)(cid:12)(cid:12) ∂ ξξ p T p T − (cid:16) ∂ ξ p T p T (cid:17) + 2 ξ ∂ ξ p T p T + ηξ (cid:12)(cid:12)(cid:12)(cid:12) ≤ ce − bT or some positive constant c. The long-term sensitivities with respect to the parameters b, a and σ are described as (cid:12)(cid:12)(cid:12)(cid:12) T ∂ b p T p T + η (cid:12)(cid:12)(cid:12)(cid:12) ≤ cT (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T ∂ a p T p T − bσ p ( a + σ / + 2 qσ − ( a + σ / p ( a + σ / + 2 qσ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ cT (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T ∂ σ p T p T + b ( a + σ / q − p ( a + σ / + 2 qσ ) σ p ( a + σ / + 2 qσ − bσ ( p ( a + σ / + 2 qσ − a − σ / a (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ cT for some positive constant c. For the proofs of these asymptotic behaviors, see Appendix B.
In this section, we consider the CEV model dX t X t = ( µ − θX βt ) dt + σX tβ dB t , X = ξ for β, µ, ξ > , σ = 0 , θ ≥ . As is well known, the process X stays positive, thus we put thedomain D = (0 , ∞ ) . When θ = 0 , the process X is the standard CEV model. The expectation p T of our interest is p T := E P [ e − q R T X − βt dt ]for q > . The process X and the expectation p T are specified by the quadruple of functions(( µ − θx β ) x, σx β +1 , qx − β , , x ∈ D . The CEV model can be transformed into the CIR model by defining Y t = X − βt . By the Itoformula, dY t = ( b − aY t ) dt + Σ p Y t dB t , Y = ξ − β where b = 2 βθ + β (2 β + 1) σ , a = 2 βµ , Σ = − βσ . Note that b > Σ / p T is p T = E P [ e − q R T Y t dt ] . The sensitivities of the expectation p T for the CIR model Y have already been analyzed inSection 5.1, thus we use the previous results. The corresponding recurrent eigenpairs are( λ, φ ( x )) = (cid:0) θη + (cid:0) β + 12 (cid:1) σ η, e − η β x − β (cid:1) , (ˆ λ, ˆ φ ( x )) = (˜ λ, ˜ φ ( x )) = (2 β p µ + 2 qσ , η := √ µ +2 qσ − µσ . For the long-term first-order and second-order sensitivities with respect to the initial-value,we have (cid:12)(cid:12)(cid:12)(cid:12) ∂ ξ p T p T − βηξ − β − (cid:12)(cid:12)(cid:12)(cid:12) ≤ ce − β √ µ +2 qσ T , (cid:12)(cid:12)(cid:12)(cid:12) ∂ ξξ p T p T − βη (2 βηξ − β − β − ξ − β − (cid:12)(cid:12)(cid:12)(cid:12) ≤ ce − β √ µ +2 qσ T for some positive constant c. We can also provide a higher-order convergence rate of the second-order sensitivity as (cid:12)(cid:12)(cid:12)(cid:12) ∂ ξξ p T p T − (cid:16) ∂ ξ p T p T (cid:17) + 2 βη (2 β + 1) ξ − β − (cid:12)(cid:12)(cid:12)(cid:12) ≤ ce − β √ µ +2 qσ T . he long-term sensitivities with respect to the parameters µ, θ, σ and β are described as (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T ∂ µ p T p T − (cid:16) θσ + β + 12 (cid:17) p µ + 2 qσ − µ p µ + 2 qσ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ cT (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T ∂ θ p T p T + p µ + 2 qσ − µσ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ cT (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T ∂ σ p T p T + qσ (2 β + 1) − θ ( µ + qσ ) σ p µ + 2 qσ + 2 θµσ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ cT (cid:12)(cid:12)(cid:12)(cid:12) T ∂ β p T p T + p µ + 2 qσ − µ (cid:12)(cid:12)(cid:12)(cid:12) ≤ cT for some positive constant c. We consider the CEV model dX t X t = ( µ − θX βt ) dt + σX tβ dB t , X = ξ for β, µ, ξ > , σ = 0 , θ ≥ . Since the process X stays positive, we put the domain D = (0 , ∞ ) . The expectation p T of our interest is p T := E P [ e − q R T X βt dt ]for q > . The process X and the expectation p T are specified by the quadruple of functions(( µ − θx β ) x, σx β +1 , qx β , , x ∈ D . The CEV model can be transformed into the 3 / Y t = X βt . By the Itoformula, dY t = ( b − aY t ) Y t dt + Σ Y / t dB t , Y = ξ β where b = 2 βµ , a = 2 βθ − β (2 β − σ , Σ = 2 βσ .
Note that a > − Σ / . The expectation p T is p T = E P [ e − q R T Y t dt ] . The sensitivities of the expectation p T for the 3 / Y have already been analyzed inSection 5.2, thus we use the previous results. The corresponding recurrent eigenpairs are( λ, φ ( x )) = ( µη, x − βη ) , (ˆ λ, ˆ φ ( x )) = (˜ λ, ˜ φ ( x )) = (2 βµ, x − β )where η := p ( θ + σ / + 2 qσ − ( θ + σ / σ . For the long-term first-order and second-order sensitivities with respect to the initial-value,we have (cid:12)(cid:12)(cid:12)(cid:12) ∂ ξ p T p T + 2 βηξ (cid:12)(cid:12)(cid:12)(cid:12) ≤ ce − βµT , (cid:12)(cid:12)(cid:12)(cid:12) ∂ ξξ p T p T − βη (2 βη + 1) ξ (cid:12)(cid:12)(cid:12)(cid:12) ≤ ce − βµT for some positive constant c. We can also provide a higher-order convergence rate of the second-order sensitivity as (cid:12)(cid:12)(cid:12)(cid:12) ∂ ξξ p T p T − (cid:16) ∂ ξ p T p T (cid:17) + 4 βξ ∂ ξ p T p T + 2 βη (4 β − ξ (cid:12)(cid:12)(cid:12)(cid:12) ≤ ce − βµT . he long-term sensitivities with respect to the parameters µ, θ, σ and β are described as (cid:12)(cid:12)(cid:12)(cid:12) T ∂ µ p T p T + η (cid:12)(cid:12)(cid:12)(cid:12) ≤ cT (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T ∂ θ p T p T − µσ p ( θ + σ / + 2 qσ − ( θ + σ / p ( θ + σ / + 2 qσ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ cT (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T ∂ σ p T p T + µ ( θ + σ / q − p ( θ + σ / + 2 qσ ) σ p ( θ + σ / + 2 qσ − µσ ( p ( θ + σ / + 2 qσ − θ − σ / θ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ cT (cid:12)(cid:12)(cid:12)(cid:12) ∂ β p T p T (cid:12)(cid:12)(cid:12)(cid:12) ≤ c for some positive constant c. This paper investigated the large-time asymptotic behavior of the sensitivities of cash flows.The price of cash flows is given in expectation form as p T = E P ξ [ e − R T r ( X s ) ds h ( X T )] . (6.1)We studied the extent to which this expectation is affected by small changes of the underlyingMarkov diffusion X. The main idea is a PDE representation of the expectation by incorporat-ing the Hansen–Scheinkman decomposition method. The sensitivities of long-term cash flowsand their large-time convergence rates can be represented via simple expressions in terms ofeigenvalues and eigenfunctions of the pricing operator h E P ξ [ e − R T r ( X s ) ds h ( X T )].Essentially, we demonstrated two types of long-term sensitivities. First, the first-order andsecond-order sensitivities with respect to the initial value ξ = X were investigated. Using theHansen–Scheinkman decomposition, we can express the expectation p T as p T = φ ( ξ ) e − λT f ( T, ξ )with recurrent eigenpair ( λ, φ ) and remainder function f ( T, ξ ) . Applying the Hansen–Scheinkmandecomposition repeatedly, the derivative f x ( T, ξ ) has the decomposition f x ( T, ξ ) = ˆ φ ( ξ ) e − ˆ λT ˆ f ( T, ξ )with recurrent eigenpair (ˆ λ, ˆ φ ) and remainder function ˆ f ( T, ξ ) . Under appropriate conditions,the first-order sensitivity and its convergence rate with respect to the initial value are given by (cid:12)(cid:12)(cid:12)(cid:12) ∂ ξ p T p T − φ ′ ( ξ ) φ ( ξ ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ ce − ˆ λT , T ≥ c, which is independent of T. For the second-order sensitivity respectto the initial value, a similar expression is obtained. We have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ ξξ p T p T − (cid:16) ∂ ξ p T p T (cid:17) − φ ′′ ( ξ ) φ ( ξ ) + (cid:16) φ ′ ( ξ ) φ ( ξ ) (cid:17) − ˆ φ ′ ( ξ )ˆ φ ( ξ ) ∂ ξ p T p T + ˆ φ ′ ( ξ )ˆ φ ( ξ ) φ ′ ( ξ ) φ ( ξ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ c ( e − ˜ λT + e − ˆ λT ) e − ˆ λT for some positive constant c, which is independent of T, where ˜ λ is a recurrent eigenvalue.Second, the sensitivities with respect to the drift and diffusion terms were demonstrated.From the Hansen–Scheinkman decomposition, the perturbed expectation p ǫT = E P ξ [ e − R T r ( X ǫs ) ds f ( X ǫT )]induced by the perturbed process X ǫ is expressed as ( ǫ ) T = φ ( ǫ ) ( ξ ) e − λ ( ǫ ) T f ( ǫ ) ( T, ξ )with recurrent eigenpair ( λ ( ǫ ) , φ ( ǫ ) ) and remainder function f ( ǫ ) ( T, ξ ) . The long-term sensitivityof p ( ǫ ) T with respect to the perturbation parameter ǫ can be expressed in a simple form as (cid:12)(cid:12)(cid:12)(cid:12) T ∂∂ǫ (cid:12)(cid:12)(cid:12) ǫ =0 ln p ( ǫ ) T + ∂∂ǫ (cid:12)(cid:12)(cid:12) ǫ =0 λ ( ǫ ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ cT for some positive constant c, which is independent of T. We presented applications of these results to three practical problems: utility maximization,entropic risk measures and bond prices. Under factor models and local volatility models, theseproblems can be transformed into the expectation form in Eq.(6.1). As specific examples, ex-plicit formulas for several market models, namely the CIR model, the 3/2 model and the CEVmodel, were investigated.
Acknowledgement.
This research was supported by the National Research Foundation of Korea (NRF) grant fundedby the Ministry of Science and ICT (No. 2018R1C1B5085491 and No. 2017R1A5A1015626)and the Ministry of Education (No. 2019R1A6A1A10073437) through Basic Science ResearchProgram.
A CIR model
Let (Ω , F , ( F t ) t ≥ , ( P t ) t ≥ ) be a consistent probability space that has a one-dimensional Brown-ian motion B = ( B t ) t ≥ . The filtration ( F t ) t ≥ is the completed filtration generated by B. TheCIR model is a process given as a solution of dX t = ( b − aX t ) dt + σ p X t dB t , X = ξ for a, σ, ξ > b > σ . For q > h with polynomialgrowth, we define p T = E P [ e − q R T X s ds h ( X T )] . It can be shown that the quadruple of functions( b − ax, σ √ x, qx, h ( x )) , x > λ, φ ( x )) := ( bη, e − ηx )where α := p a + 2 qσ , η := α − aσ . Under the consistent family of recurrent eigen-measures (ˆ P t ) t ≥ , the processˆ B t = ση Z t p X s ds + B t , t ≥ X follows dX t = ( b − αX t ) dt + σ p X t d ˆ B t . sing the Hansen–Scheinkman decomposition, we have p T = E P ξ [ e − q R T X s ds h ( X T )] = E ˆ P ξ [ h ( X T ) e ηX T ] e − ηξ e − λT . (A.1)For t ∈ [0 , ∞ ) and x > , the remainder function is f ( t, x ) = E ˆ P x [ h ( X t ) e ηX t ] (A.2)so that p T = f ( T, ξ ) e − ηξ e − λT . For nonzero and nonnegative h with polynomial growth, it iseasy to show that f ( T, ξ ) converges to a positive constant as T → ∞ by using Lemma A.1. It isalso easy to check that f is C , by considering the density function of X t . We will investigatethe behavior of the function f ( T, ξ ) by expressing this function as a solution of a second-orderdifferential equation. Using the Feynman–Kac formula, the function f satisfies − f t + 12 σ xf xx + ( b − αx ) f x = 0 , f (0 , x ) = h ( x ) e ηx . (A.3) Lemma A.1.
Let ˆ B be a Brownian motion on the consistent probability space (Ω , F , ( F t ) t ≥ , (ˆ P t ) t ≥ ) . Suppose that X is a solution of dX t = ( b − αX t ) dt + σ p X t d ˆ B t , X = ξ where α, σ, ξ > and b > σ . Then, for β < α/σ , we have E ˆ P [ e βX T ] = (cid:16) − βc ( T ) (cid:17) b/σ e β − βc ( T ) e − αT ξ where c ( T ) := σ (1 − e − αT ) / α. Thus, in this case, E ˆ P [ e βX T ] ≤ (cid:16) − βσ / α (cid:17) b/σ e β − βσ / α e − αT ξ , and lim T →∞ E ˆ P [ e βX T ] = (cid:16) − βσ / α (cid:17) b/σ . Refer to (Jeanblanc et al., 2009, Corollary 6.3.4.4) for the proof.
A.1 First-order sensitivity of ξ We estimate the large-time asymptotic behavior of the first-order sensitivity of p T with respectto the initial value ξ. In this section, assume that h is continuously differentiable and that h and h ′ have polynomial growth. From Eq.(A.1), it follows that ∂ ξ p T p T = f x ( T, ξ ) f ( T, ξ ) − η . (A.4)The function f x ( t, x ) satisfies − f xt + 12 σ xf xxx + (cid:16) b + 12 σ − αx (cid:17) f xx − αf x = 0 , f x (0 , x ) = ( h ′ ( x ) + ηh ( x )) e ηx , (A.5)which is obtained from Eq.(A.3) by taking the differentiation in x. Note that since f is C , and every coefficient is continuously differentiable in x in Eq.(A.3), the function f is thricecontinuously differentiable in x. It is easy to show that the quadruple of functions( b + σ / − αx, σ √ x, α, ( h ′ + ηh ) e ηx ) , x > X is the solution of d ˆ X t = ( b + σ / − α ˆ X t ) dt + σ ˆ X / t d ˆ B t . emma A.2. The remainder function f satisfies f x ( t, x ) = E ˆ P [( h ′ ( ˆ X t ) + ηh ( ˆ X t )) e η ˆ X t | ˆ X = x ] e − αt (A.6) for x > and t ≥ . Proof.
Define g ( t, x ) := f x ( t, x ) φ ( x ) . Eq.(A.5) gives − g t + 12 σ xg xx + ( b + σ / − ax ) g x + (cid:16) −
12 ( α + a ) ηx + ( b + σ / η − α (cid:17) g = 0 g (0 , x ) = h ′ ( x ) + ηh ( x ) . Consider a consistent family ( Q t ) t ≥ of probability measures where each Q t is a probabilitymeasure on F t defined as d Q t d ˆ P t = φ ( ˆ X ) φ ( ˆ X t ) e R t L φ ( ˆ Xs ) φ ( ˆ Xs ) ds = e η ( ˆ X t − x )+ R t ( α + a ) η ˆ X s − ( b + σ / η ds = e − σ η R t ˆ X s ds + ση R t ˆ X / s d ˆ B s . It is easy to check that ˆ X satisfies d ˆ X t = ( b + σ / − α ˆ X t ) dt + σ ˆ X / t d ˆ B t = ( b + σ / − a ˆ X t ) dt + σ ˆ X / t dB Q t for a ( Q t ) t ≥ -Brownian motion ( B Q t ) t ≥ . By (Pinsky, 1995, Theorem 5.1.8), since this processdoes not reach the boundaries under the consistent family of probability measures ( Q t ) t ≥ , the Q T -local martingale ( e − σ η R t ˆ X s ds + ση R t ˆ X / s d ˆ B s ) ≤ t ≤ T is a Q T -martingale. Observe that theoperator ˆ L given as ˆ L f = 12 σ xf ′′ ( x ) + ( b + σ / − ax ) f ′ ( x )is the infinitesimal generator of ˆ X under the consistent family of probability measures ( Q t ) T ≥ , and for φ ( x ) = e − ηx , we getˆ L φ ( x ) = (cid:0)
12 ( α + a ) ηx − ( b + σ / η (cid:1) e − ηx . The Feynman–Kac formula (Remark 3.3 or Proposition 3.1) gives that f x ( t, x ) φ ( x ) = g ( t, x ) = E Q (cid:2) e R t − ( α + a ) η ˆ X s +( b + σ / η − α ds ( h ′ ( ˆ X t ) + ηh ( ˆ X t )) (cid:12)(cid:12) ˆ X = x (cid:3) = E Q h e − R t L φφ ( ˆ X s ) ds φ ( ˆ X t ) φ ( ˆ X ) e R t − α ds ( h ′ ( ˆ X t ) + ηh ( ˆ X t )) e η ˆ X t (cid:12)(cid:12)(cid:12) ˆ X = x i φ ( x )= E Q h d ˆ P t d Q t e − αt ( h ′ ( ˆ X t ) + ηh ( ˆ X t )) e η ˆ X t (cid:12)(cid:12)(cid:12) ˆ X = x i φ ( x )= E ˆ P (cid:2) ( h ′ ( ˆ X t ) + ηh ( ˆ X t )) e η ˆ X t (cid:12)(cid:12) ˆ X = x (cid:3) φ ( x ) e − αt , which implies that f x ( t, x ) = E ˆ P (cid:2) ( h ′ ( ˆ X t ) + ηh ( ˆ X t )) e η ˆ X t (cid:12)(cid:12) ˆ X = x (cid:3) e − αt . Condition (iv) of Proposition 3.1 can be confirmed from Lemma A.1 and the density functionof X t , and the other conditions are trivial. sing Eq.(A.6), we can obtain the large-time behavior of ∂ ξ p T . Since h and h ′ have poly-nomial growth, for η < β < α/σ , there is a positive constant c = c ( β ) such that | h ′ ( x ) + ηh ( x ) | e ηx ≤ c e βx for x > . From Lemma A.1, we have | f x ( t, x ) | e αt ≤ E ˆ P h | h ′ ( ˆ X t ) + ηh ( ˆ X t ) | e η ˆ X t (cid:12)(cid:12)(cid:12) ˆ X = x i ≤ c E ˆ P [ e β ˆ X t | ˆ X = x ] ≤ c (cid:16) − βσ / α (cid:17) b/σ e β − βσ / α e − αt x ≤ c (A.7)for some positive constant c which depends on x but does not depend on t. It follows that | f x ( t, x ) | ≤ c e − αt . (A.8)Therefore, (cid:12)(cid:12)(cid:12)(cid:12) ∂ ξ p T p T + η (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) f x ( T, ξ ) f ( T, ξ ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ c e − αT for some positive constant c . This gives the desired result.
A.2 Second-order sensitivity of ξ We analyze the second-order sensitivity with respect to the initial value ξ. In this section,assume that h is twice continuously differentiable and that h, h ′ , h ′′ have polynomial growth.From Eq.(A.4), we know that ∂ ξξ p T p T = f xx ( T, ξ ) f ( T, ξ ) − η f x ( T, ξ ) f ( T, ξ ) + η . (A.9)Since we already estimated the large-time asymptotic behavior of f x ( T, ξ ) , it suffices to investi-gate the second-order derivative f xx ( T, ξ ) . Define ˜ P := ˆ P (to be consistent with the notations inTable 1) and ˆ f ( t, x ) := E ˜ P [( h ′ ( ˆ X t ) + ηh ( ˆ X t )) e η ˆ X t | ˆ X = x ]so that f x ( t, x ) = ˆ f ( t, x ) e − αt , which gives f xx ( t, x ) = ˆ f x ( t, x ) e − αt . By the Feynman–Kac formula, we have − ˆ f t + 12 σ x ˆ f xx + ( b + σ / − αx ) ˆ f x = 0 , ˆ f (0 , x ) = ( h ′ ( x ) + ηh ( x )) e ηx . Since ˆ f is C , and every coefficient is continuously differentiable in x , the function ˆ f is thricecontinuously differentiable in x. Differentiate this PDE in x, then − ˆ f xt + 12 σ x ˆ f xxx + ( b + σ − αx ) ˆ f xx − α ˆ f x = 0 , ˆ f x (0 , x ) = ( h ′′ ( x ) + 2 ηh ′ ( x ) + η h ( x )) e ηx . It is easy to show that the quadruple of functions( b + σ − αx, σ √ x, α, ( h ′′ ( x ) + 2 ηh ′ ( x ) + η h ( x )) e ηx ) , x > X is a solution of d ˜ X t = ( b + σ − α ˜ X t ) dt + σ ˜ X / t d ˜ B t . e observe that ˆ f x ( t, x ) satisfiesˆ f x ( t, x ) = E ˜ P [( h ′′ ( ˜ X t ) + 2 ηh ′ ( ˜ X t ) + η h ( ˜ X t )) e η ˜ X t | ˜ X = x ] e − αt . This is directly obtained from Proposition 3.1 by the same argument used in the derivation ofEq.(A.6).To analyze ˆ f x ( t, x ) , we apply the same argument used in Eq.(A.7) and Eq.(A.8). For η < β < α/σ , there is a positive constant c = c ( β ) such that | ( h ′′ ( x ) + 2 ηh ′ ( x ) + η h ( x )) | e ηx ≤ c e βx for x > . Using Lemma A.1, we have | ˆ f x ( t, x ) | e αt ≤ E ˜ P h | h ′′ ( ˜ X t ) + 2 ηh ′ ( ˜ X t ) + η h ( ˜ X t ) | e η ˜ X t (cid:12)(cid:12)(cid:12) ˜ X = x i ≤ c E ˜ P [ e β ˜ X t | ˜ X = x ] ≤ c (cid:16) − βσ / α (cid:17) b/σ e β − βσ / α e − αt x ≤ c (A.10)for some positive constant c which depends on x but does not depend on t. It follows that | ˆ f x ( t, x ) | ≤ c e − αt , which gives | f xx ( t, x ) | = | ˆ f x ( t, x ) | e − αt ≤ c e − αt . (A.11)Therefore, (cid:12)(cid:12)(cid:12)(cid:12) ∂ ξξ p T p T − η (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) f ξξ ( T, ξ ) f ( T, ξ ) (cid:12)(cid:12)(cid:12)(cid:12) + 2 η (cid:12)(cid:12)(cid:12)(cid:12) f ξ ( T, ξ ) f ( T, ξ ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ c e − αT for some positive constant c . We can also provide a higher-order convergence rate as follows.From Eq.(A.4) and Eq.(A.9), (cid:12)(cid:12)(cid:12)(cid:12) ∂ ξξ p T p T − (cid:16) ∂ ξ p T p T (cid:17) (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) f ξξ ( T, ξ ) f ( T, ξ ) (cid:12)(cid:12)(cid:12)(cid:12) + (cid:18) f ξ ( T, ξ ) f ( T, ξ ) (cid:19) ≤ c e − αT for some positive constant c . A.3 Sensitivity of b We investigate the large-time asymptotic behavior of the sensitivity with respect to the pa-rameter b. In this section, assume that h is continuously differentiable and that h and h ′ havepolynomial growth. Using Eq.(A.1) and Eq.(A.2), since η is independent of b and λ = bη, wehave ∂ b p T p T = f b ( T, ξ ) f ( T, ξ ) − ηT . It can be easily shown that f is continuously differentiable in b by considering the densityfunction of X t or by using (Park, 2018, Theorem 4.8). We focus on the large-time behavior of f b ( T, ξ ) . Differentiate Eq.(A.3) in b, then − f bt + 12 σ xf bxx + ( b − αx ) f bx + f x = 0 , f b (0 , x ) = 0 . From the Feynman–Kac formula, one can show that f b ( t, x ) = E ˆ P x h Z t f x ( t − s, X s ) ds i y the same method used in the proof of Lemma A.2.We can estimate the expectation on the right-hand side by using the same method in SectionA.1. For η < β < α/σ , Eq.(A.6) and Eq.(A.7) implies that | f x ( t − s, x ) | ≤ c (cid:16) − βσ / α (cid:17) b/σ e β − βσ / α e − α ( t − s ) x e − α ( t − s ) ≤ c e γx e − α ( t − s ) (A.12)where c := c (cid:16) − βσ / α (cid:17) b/σ , γ := β − βσ / α . Then, | f b ( t, x ) | ≤ E ˆ P x h Z t | f x ( t − s, X s ) | ds i = Z t E ˆ P x | f x ( t − s, X s ) | ds ≤ c Z t e − α ( t − s ) E ˆ P x [ e γX s ] ds . Since γ < α/σ , by Lemma A.1, the expectation E ˆ P x [ e γX s ] is bounded in s on [0 , ∞ ) . Thus,there is a positive constant c such that E ˆ P x [ e γX s ] ≤ c α/c , which gives | f b ( t, x ) | ≤ c α Z t e − α ( t − s ) ds = c (1 − e − αt ) . Since f ( T, ξ ) converges to a positive constant as T → ∞ , we conclude that (cid:12)(cid:12)(cid:12)(cid:12) T ∂ b p T p T + η (cid:12)(cid:12)(cid:12)(cid:12) = 1 T (cid:12)(cid:12)(cid:12)(cid:12) f b ( T, ξ ) f ( T, ξ ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ c T for some positive constant c . A.4 Sensitivity of a We investigate the large-time asymptotic behavior of the sensitivity of p T with respect to theparameter a. In this section, assume that h is continuously differentiable and that h and h ′ havepolynomial growth. From Eq.(A.1) and Eq.(A.2), it follows that ∂ a p T p T = f a ( T, ξ ) f ( T, ξ ) − ξ∂ a η − T ∂ a λ . It can be easily shown that f is continuously differentiable in a by considering the densityfunction of X t or by using (Park, 2018, Theorem 4.8). We focus on the large-time behavior of f a ( T, ξ ) . Differentiate Eq.(A.3) in a, then − f at + 12 σ xf axx + ( b − αx ) f ax − aα xf x = 0 , f a (0 , x ) = − ηα h ( x ) xe ηx . Here, we used α = p a + 2 qσ and η = α − aσ . By the Feynman–Kac formula in Remark 3.3, itfollows that f a ( t, x ) = E ˆ P x h − aα Z t X s f x ( t − s, X s ) ds − ηα h ( X t ) X t e ηX t i = − aα Z t E ˆ P x [ X s f x ( t − s, X s )] ds − ηα E ˆ P x [ h ( X t ) X t e ηX t ] . (A.13)Note that Remark 3.3 cannot be applied directly because the two terms − aα xf x and f a (0 , x ) = − ηα h ( x ) xe ηx neither have polynomial growth nor are nonnegative. To overcome this problem,define g ( t, x ) = − f a ( t, x ) , then − f t + 12 σ xg xx + ( b − αx ) g x + aα xf x = 0 , g a (0 , x ) = ηα h ( x ) xe ηx . ince this equation satisfies the hypothesis of Remark 3.3, we obtain Eq.(A.13) by the samemethod used in the proof of Lemma A.2.Now, we use Eq.(A.13) to estimate f a ( t, x ) . From Eq.(A.12), we know that | xf x ( t − s, x ) | ≤ c e δx e − α ( t − s ) (A.14)for some positive constants c and γ < δ < α/σ . By Lemma A.1, the expectation E ˆ P x [ e δX s ] isbounded in s on [0 , ∞ ) . Thus, there is a positive constant c such that E ˆ P x [ e δX s ] ≤ c α/c , whichgives Z t E ˆ P x | X s f x ( t − s, X s ) | ds ≤ c Z t e − α ( t − s ) E ˆ P x [ e δX s ] ds ≤ c α Z t e − α ( t − s ) ds = c (1 − e − αt ) ≤ c . (A.15)The expectation E ˆ P x [ h ( X t ) X t e ηX t ] is also bounded in t on [0 , ∞ ) . We conclude that (cid:12)(cid:12)(cid:12)(cid:12) T ∂ a p T p T + ∂ a λ (cid:12)(cid:12)(cid:12)(cid:12) = 1 T (cid:12)(cid:12)(cid:12)(cid:12) f a ( T, ξ ) f ( T, ξ ) − ξ∂ a η (cid:12)(cid:12)(cid:12)(cid:12) ≤ c T for some positive constant c . Direct calculation gives ∂ a λ = bσ ( aα − . A.5 Sensitivity of σ We study the large-time asymptotic behavior of the sensitivity of p T with respect to the param-eter σ. In this section, assume that h is twice continuously differentiable and that h, h ′ , h ′′ havepolynomial growth. From Eq.(A.1) and Eq.(A.2), it follows that ∂ σ p T p T = f σ ( T, ξ ) f ( T, ξ ) − ξ∂ σ η − T ∂ σ λ . It can be easily shown that f is continuously differentiable in σ by considering the densityfunction of X t or by using (Park, 2018, Theorem 4.13). We focus on the large-time behavior of f σ ( T, ξ ) . Differentiate Eq.(A.3) in σ, then − f σt + 12 σ xf σxx + ( b − αx ) f σx + σxf xx − qσα xf x = 0 , f σ (0 , x ) = h ( x ) xe ηx ∂ σ η . By the same method used in the proof of Lemma A.2, we have f σ ( t, x ) = E ˆ P x h σ Z t X s f xx ( t − s, X s ) ds − qσα Z t X s f x ( t − s, X s ) ds + h ( X t ) X t e ηX t ∂ σ η i = σ Z t E ˆ P x [ X s f xx ( t − s, X s )] ds − qσα Z t E ˆ P x [ X s f x ( t − s, X s )] ds + ∂ σ η E ˆ P x [ h ( X t ) X t e ηX t ] . We claim that | f σ ( t, x ) | is bounded in t on [0 , ∞ ) by estimating the three terms on theright-hand side. From Eq.(A.10) and Eq.(A.11), for η < β < α/σ , | f xx ( t − s, x ) | ≤ c (cid:16) − βσ / α (cid:17) b/σ e β − βσ / α x e − α ( t − s ) . For δ with β − βσ / α < δ < α/σ , there is a positive constant c such that | xf xx ( t − s, x ) | ≤ c e δx e − α ( t − s ) . By the same analysis used in Eq.(A.14) and Eq.(A.15), it follows that Z t E ˆ P x [ X s f xx ( t − s, X s )] ds s bounded in t on [0 , ∞ ) . By Eq.(A.15), the integral Z t E ˆ P x [ X s f x ( t − s, X s )] ds is bounded in t on [0 , ∞ ) . The expectation E ˆ P x [ h ( X t ) X t e ηX t ] is also bounded in t on [0 , ∞ ) . Therefore, we have | f σ ( t, x ) | ≤ c for some positive constant c . In conclusion, (cid:12)(cid:12)(cid:12)(cid:12) T ∂ σ p T p T + ∂ σ λ (cid:12)(cid:12)(cid:12)(cid:12) = 1 T (cid:12)(cid:12)(cid:12)(cid:12) f σ ( T, ξ ) f ( T, ξ ) − ξ∂ σ η (cid:12)(cid:12)(cid:12)(cid:12) ≤ c T for some positive constant c . Direct calculation gives ∂ σ λ = 2 b ( qασ − α − aσ ) . B 3/2 model
The 3 / dX t = ( b − aX t ) X t dt + σX t / dB t , X = ξ for b, σ, ξ > a > − σ / . For q > h with lineargrowth at most, we define p T = E P [ e − q R T X s ds h ( X T )] . It can be shown that the quadruple of functions(( b − ax ) x, σx / , qx, h ) , x > λ, φ ( x )) := ( bη, x − η )where η := p ( a + σ / + 2 qσ − ( a + σ / σ . Under the consistent family of recurrent eigen-measures (ˆ P t ) t ≥ , the processˆ B t = ση Z t p X s ds + B t , t ≥ X follows dX t = ( b − αX t ) X t dt + σX t / d ˆ B t where α := a + σ η. Using this consistent family of recurrent eigen-measures, we have the Hansen–Scheinkmandecomposition p T = E P ξ [ e − q R T X s ds h ( X T )] = E ˆ P ξ [ h ( X T ) X ηT ] ξ − η e − λT . (B.1)For t ∈ [0 , ∞ ) and x > , we define f ( t, x ) = E ˆ P x [ h ( X t ) X ηt ] (B.2)so that p T = f ( T, ξ ) φ ( ξ ) e − λT = f ( T, ξ ) ξ − η e − λT . (B.3) or nonzero, nonnegative Borel function h with linear growth at most, it is easy to show that f ( T, ξ ) converges to a positive constant as T → ∞ by Lemma B.1. We will investigate the large-time behavior of the function f ( T, ξ ) by expressing this function as a solution of a second-orderdifferential equation. By the Feynman–Kac formula, f satisfies − f t + 12 σ x f xx + ( b − αx ) xf x = 0 , f (0 , x ) = h ( x ) x η . (B.4) Lemma B.1.
Let ˆ B = ( ˆ B t ) t ≥ be a Brownian motion on the consistent probability space (Ω , F , ( F t ) t ≥ , (ˆ P t ) t ≥ ) . Suppose that X is a solution of dX t = ( b − αX t ) X t dt + σX t / d ˆ B t , X = ξ where α, b, σ, ξ > . Then, for
A < ασ + 2 , we have E ξ ( X AT ) = Γ( ασ + 2 − A )Γ( ασ + 2) (cid:16) bσ − e − bT (cid:17) A F (cid:16) A, ασ + 2 , − bσ e bT − ξ (cid:17) , and the expectation E ξ ( X AT ) converges to Γ( ασ + 2 − A )Γ( ασ + 2) (cid:16) bσ (cid:17) A as T → ∞ where F is the confluent hypergeometric function. Moreover, if < A < ασ + 2 , then the map H : [0 , ∞ ) × (0 , ∞ ) → R defined by H ( t, x ) = E x ( X At ) is uniformly bounded onthe domain [0 , ∞ ) × (0 , ∞ ) . Proof.
Define a process Y as Y = 1 /X, then dY t = ( θ − bY t ) dt − σ p Y t d ˆ B t , Y = ζ , where θ := α + σ and ζ := 1 /ξ. Since θ > σ / , the Feller condition is satisfied. From(Hurd and Kuznetsov, 2008, Theorem 3.1) or (Dereich et al., 2011, Section 3), we have for A < θσ = ασ + 2 , E ( X At ) = E ( Y − At ) = Γ( θσ − A )Γ( θσ ) (cid:16) bσ − e − bt (cid:17) A F (cid:16) A, θσ , − bσ ζe bt − (cid:17) . Since the confluent hypergeometric function F satisfieslim t →∞ F ( A, θσ , − bσ ζe bt − , we obtain the desired result. Moreover, if A and ασ + 2 are positive, the function z F ( A, ασ +2 , z ) is uniformly bounded in z on (0 , ∞ ) , which is the direct result from (Abramowitz and Stegun,1965, 13.1.5 on page 504). This means that the map H is uniformly bounded on the domain[0 , ∞ ) × (0 , ∞ ) . B.1 First-order sensitivity of ξ We estimate the large-time asymptotic behavior of the first-order sensitivity of p T with respectto the initial value ξ. In this section, assume that h is continuously differentiable and that h and h ′ have linear growth at most and are nonnegative (we are mainly interested in the case h = 1).From Eq.(B.1), it follows that ∂ ξ p T p T = f x ( T, ξ ) f ( T, ξ ) − ηξ . e focus on the term f x ( T, ξ ) . Since f is C , and every coefficient is continuously differentiablein x in Eq.(B.4), the function f is thrice continuously differentiable in x. This gives − f xt + 12 σ x f xxx + ( b − ( α − σ ) x ) xf xx + ( b − αx ) f x = 0 , f x (0 , x ) = ( xh ′ ( x ) + ηh ( x )) x η − . (B.5)It is easy to show that the quadruple of functions(( b − ( α − σ ) x ) x, σx / , − b + 2 αx, ( xh ′ ( x ) + ηh ( x )) x η − ) , x > X is the solution of d ˆ X t = ( b − ( α − σ ) ˆ X t ) ˆ X t dt + σ ˆ X / t d ˆ B t . We show that the remainder function f satisfies f x ( t, x ) = E ˆ P [( ˆ X t h ′ ( ˆ X t ) + ηh ( ˆ X t )) ˆ X η − t e − α R t ˆ X s ds | ˆ X = x ] e bt . (B.7)To achieve this equality, we use the Feynman–Kac formula in Remark 3.3. However, Remark3.3 cannot be applied directly because the volatility function σx / of ˆ X does not have lineargrowth. Instead, we define Y := 1 / ˆ X and g ( t, y ) := f x ( t, /y ) . Then, Y is a CIR process dY t = (( α − σ / − bY t ) dt − σ p Y t d ˆ B t , and Eq.(B.5) becomes − g t + 12 σ yg yy + (( α − σ / − by ) g y + ( b − α/y ) g = 0 , g (0 , y ) = ( y − h ′ ( y − ) + ηh ( y − )) y − η . Now, the quadruple of functions (cid:16) ( α − σ / − by, − σ √ y, − b + 2 α/y, ( y − h ′ ( y − ) + ηh ( y − )) y − η (cid:17) , y > ≤ t ≤ T | f ( t, /y ) | (thus, max ≤ t ≤ T | g ( t, y ) | = max ≤ t ≤ T | f x ( t, /y ) | by the Schauder estimation) is bounded in y by Lemma B.1. It follows that g ( t, y ) = E ˆ P [( Y − t h ′ ( Y − t ) + ηh ( Y − t )) Y − ηt e − α R t Y − t ds | Y = y ] e bt = E ˆ P [( ˆ X t h ′ ( ˆ X t ) + ηh ( ˆ X t )) ˆ X η − t e − α R t ˆ X s ds | ˆ X = 1 /y ] e bt . Thus, f x ( t, x ) = g ( t, /x ) = E ˆ P [( ˆ X t h ′ ( ˆ X t ) + ηh ( ˆ X t )) ˆ X η − t e − α R t ˆ X s ds | ˆ X = x ] e bt , which is the desired result.To analyze f x ( t, x ) e − bt = E ˆ P [( ˆ X t h ′ ( ˆ X t ) + ηh ( ˆ X t )) ˆ X η − t e − α R t ˆ X s ds | ˆ X = x ] , we apply the Hansen–Scheinkman decomposition. It can be shown that the pair (2 b, x − ) is therecurrent eigenpair by considering the generator of ˆ X with killing rate 2 α ˆ X t , which is given as L ψ ( x ) := 12 σ x ψ ′′ ( x ) + ( b − ( α − σ ) x ) xψ ′ ( x ) − αxψ ( x ) . onsider a consistent family (˜ P t ) t ≥ of probability measures where each ˜ P t is defined on F t as d ˜ P t d ˆ P t = e bt − α R t ˆ X s ds ˆ X ˆ X t . Then, ˜ B t := 2 σ Z t ˆ X / s ds + ˆ B t , t ≥ P t ) t ≥ -Brownian motion, and ˆ X follows d ˆ X t = ( b − ( α + 12 σ ) ˆ X t ) ˆ X t dt + σ ˆ X / t d ˜ B t . Using this consistent family of probability measures, we have f x ( t, x ) = E ˆ P [( ˆ X t h ′ ( ˆ X t ) + ηh ( ˆ X t )) ˆ X η − t e − α R t ˆ X s ds | ˆ X = x ] e bt = E ˜ P [( ˆ X t h ′ ( ˆ X t ) + ηh ( ˆ X t )) ˆ X η +1 t | ˆ X = x ] e − bt x − . (B.8)We can obtain the large-time behavior of ∂ ξ p T by using Eq.(B.8). Since h and h ′ have lineargrowth at most, there is a positive constant c such that | ˆ X t h ′ ( ˆ X t ) + ηh ( ˆ X t ) | ˆ X η +1 t ≤ c ˆ X η +3 t . By Lemma B.1, the expectation E ˜ P (cid:2) | ˆ X t h ′ ( ˆ X t ) + ηh ( ˆ X t ) | ˆ X η +1 t (cid:12)(cid:12) ˆ X = x (cid:3) (B.9)is uniformly bounded in ( t, x ) since the constant A in the lemma satisfies A = η + 3 < ασ + 3 . Therefore, (cid:12)(cid:12)(cid:12)(cid:12) ∂ ξ p T p T + ηξ (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) f ξ ( T, ξ ) f ( T, ξ ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ c e − bT (B.10)for some positive constant c . This gives the desired result.
B.2 Second-order sensitivity of ξ We investigate the large-time asymptotic behavior of the second-order sensitivity with respectto the initial value ξ. In this section, assume that h is twice continuously differentiable, h and h ′ have linear growth at most and h ′′ is bounded, moreover h, h ′ , h ′′ are nonnegative (we aremainly interested in the case h = 1). From Eq.(B.1), ∂ ξξ p T p T = f xx ( T, ξ ) f ( T, ξ ) − ηξ f x ( T, ξ ) f ( T, ξ ) + η ( η + 1) ξ . Since we already estimated the large-time asymptotic behavior of f x ( T, ξ ) , we focus on thesecond-order derivative f xx ( T, ξ ) . Defineˆ f ( t, x ) := E ˜ P [( ˆ X t h ′ ( ˆ X t ) + ηh ( ˆ X t )) ˆ X η +1 t | ˆ X = x ]so that Eq.(B.8) gives f x ( t, x ) = ˆ f ( t, x ) e − bt x − . Thus, f xx ( t, x ) = ˆ f x ( t, x ) e − bt x − − f ( t, x ) e − bt x − . (B.11)We need to estimate the large-time behavior of ˆ f x ( t, x ) . The Feynman–Kac formula statesthat − ˆ f t + 12 σ x ˆ f xx + ( b − ( α + 12 σ ) x ) x ˆ f x = 0 , ˆ f (0 , x ) = ( h ′ ( x ) x + ηh ( x )) x η +1 . ifferentiate this equation in x, then − ˆ f xt + 12 σ x ˆ f xxx + ( b − ( α − σ ) x ) x ˆ f xx + ( b − (2 α + σ ) x ) ˆ f x = 0 , ˆ f x (0 , x ) = ( h ′′ ( x ) x + 2( η + 1) xh ′ ( x ) + η ( η + 1) h ( x )) x η . It is easy to show that the quadruple of functions (cid:16) ( b − ( α − σ ) x ) x, σx / , − b + (2 α + σ ) x, ( h ′′ ( x ) x + 2( η + 1) xh ′ ( x ) + η ( η + 1) h ( x )) x η (cid:17) , x > X is the solution of d ˜ X t = ( b − ( α − σ ) ˜ X t ) ˜ X t dt + σ ˜ X / t d ˜ B t . We observe thatˆ f x ( t, x ) = E ˜ P [( h ′′ ( ˜ X t ) ˜ X t + 2( η + 1) ˜ X t h ′ ( ˜ X t ) + η ( η + 1) h ( ˜ X t )) ˜ X ηt e − (2 α + σ ) R t ˜ X s ds | ˜ X = x ] e bt . This can be obtained by the same argument used in the derivation of Eq.(B.7) by considering Y = 1 /X. To analyze the expectation above, we apply the Hansen–Scheinkman decomposition. It canbe shown that the pair (2 b, x − ) is the recurrent eigenpair by considering the generator of ˜ X with killing rate (2 α + σ ) ˜ X t , which is given as L ψ ( x ) := 12 σ x ψ ′′ ( x ) + ( b − ( α − σ ) x ) xψ ′ ( x ) − (2 α + σ ) xψ ( x ) . Consider a consistent family (¯ P t ) t ≥ of probability measures where each ¯ P t is defined on F t as d ¯ P t d ˜ P t = e bt − (2 α + σ ) R t ˜ X s ds ˜ X ˜ X t . Then, ¯ B t := 2 σ Z t ˜ X / s ds + ˜ B t , t ≥ P t ) t ≥ -Brownian motion, and ˜ X follows d ˜ X t = ( b − ( α + σ ) ˜ X t ) ˜ X t dt + σ ˜ X / t d ¯ B t . Using this consistent family of probability measures (¯ P t ) t ≥ , we haveˆ f x ( t, x ) = E ˜ P [( h ′′ ( ˜ X t ) ˜ X t + 2( η + 1) ˜ X t h ′ ( ˜ X t ) + η ( η + 1) h ( ˜ X t )) ˜ X ηt e − (2 α + σ ) R t ˜ X s ds | ˜ X = x ] e bt = E ¯ P [( h ′′ ( ˜ X t ) ˜ X t + 2( η + 1) ˜ X t h ′ ( ˜ X t ) + η ( η + 1) h ( ˜ X t )) ˜ X η +2 t | ˜ X = x ] e − bt x − . (B.12)Since h, h ′ have linear growth at most and h ′′ is bounded, by Lemma B.1, the expectation E ¯ P (cid:2) | h ′′ ( ˜ X t ) ˜ X t + 2( η + 1) ˜ X t h ′ ( ˜ X t ) + η ( η + 1) h ( ˜ X t ) | ˜ X η +2 t (cid:12)(cid:12) ˜ X = x (cid:3) converges to a positive constant as t → ∞ since the constant A in the lemma satisfies A = η + 4 < ασ + 4 . Thus, | f xx ( t, x ) | = | ˆ f x ( t, x ) e − bt x − − g ( t, x ) e − bt x − |≤ | ˆ f x ( t, x ) | e − bt x − + 2 | g ( t, x ) | e − bt x − ≤ c e − bt x − + c e − bt x − ≤ c e − bt or some positive constants c and c , which are independent of t. From Eq.(B.10), we concludethat (cid:12)(cid:12)(cid:12)(cid:12) ∂ ξξ p T p T − η ( η + 1) ξ (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) f xx ( T, ξ ) f ( T, ξ ) − ηξ f x ( T, ξ ) f ( T, ξ ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) f xx ( T, ξ ) f ( T, ξ ) (cid:12)(cid:12)(cid:12)(cid:12) + 2 ηξ (cid:12)(cid:12)(cid:12)(cid:12) f x ( T, ξ ) f ( T, ξ ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ c e − bT for some positive constant c . This gives the convergence rate of the second-order sensitivity.We can also provide a higher-order convergence rate of the second-order sensitivity as follows.From Eq.(B.11) and p T = f ( T, ξ ) e λT φ ( ξ ) presented in Eq.(B.3), it follows that ∂ ξξ p T p T − (cid:16) ∂ ξ p T p T (cid:17) + 2 ξ (cid:16) ∂ ξ p T p T − φ ′ ( ξ ) φ ( ξ ) (cid:17) − φ ′′ ( ξ ) φ ( ξ ) + (cid:16) φ ′ ( ξ ) φ ( ξ ) (cid:17) = g x ( T, ξ ) g ( T, ξ ) f x ( T, ξ ) f ( T, ξ ) − (cid:16) f x ( T, ξ ) f ( T, ξ ) (cid:17) . Using φ ( ξ ) = ξ − η , we have (cid:12)(cid:12)(cid:12)(cid:12) ∂ ξξ p T p T − (cid:16) ∂ ξ p T p T (cid:17) + 2 ξ ∂ ξ p T p T + ηξ (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) g x ( T, ξ ) g ( T, ξ ) f x ( T, ξ ) f ( T, ξ ) (cid:12)(cid:12)(cid:12)(cid:12) + (cid:16) f x ( T, ξ ) f ( T, ξ ) (cid:17) ≤ c e − bT for some positive constant c . For the last inequality, we used Eq.(B.10) and Eq.(B.12).
B.3 Sensitivity of b We investigate the large-time asymptotic behavior of the sensitivity of p T with respect to theparameter b. In this section, assume that h is continuously differentiable and that h, h ′ havelinear growth at most and are nonnegative. Recall the Hansen–Scheinkman decomposition p T = f ( T, ξ ) ξ − η e − λT and the remainder function f ( t, x ) = E ˆ P x [ h ( X t ) X ηt ] in Eq.(B.1) and Eq.(B.2).Since η is independent of b and λ = bη, we have ∂ b p T p T = f b ( T, ξ ) f ( T, ξ ) − ηT . It can be easily shown that f is continuously differentiable in b by considering the densityfunction of X t or by using (Park, 2018, Theorem 4.8). We focus on the large-time behavior of f b ( T, ξ ) . Differentiate Eq.(B.4) in b, then − f bt + 12 σ x f bxx + ( b − αx ) xf bx + xf x = 0 , f b (0 , x ) = 0 . (B.13)Recall that the quadruple of functions in Eq.(B.6) satisfies Assumptions 2.1 – 2.5.From this PDE, we observe that the remainder function f satisfies f b ( t, x ) = E ˆ P x h Z t X s f x ( s, X s ) ds i . However, to obtain this equality, the Feynman–Kac formula in Remark 3.3 cannot be applieddirectly because the volatility function σx / of X does not have linear growth. Instead, wedefine Y := 1 /X and g ( t, y ) := f b ( t, /y ) . Then, Y is a CIR process dY t = ( α + σ − bY t ) dt − σ p Y t d ˆ B t , and Eq.(B.13) becomes − g t + 12 σ yg yy + ( α + σ − by ) g y + (1 /y ) f x ( t, /y ) = 0 , g (0 , y ) = 0 . his PDE satisfies all the conditions in Remark 3.3. Note that the function (1 /y ) f x ( t, /y )has linear growth at most in y by Eq.(B.8) because Eq.(B.9) is uniformly bounded in ( t, x ) . Itfollows that g ( t, y ) = E ˆ P h Z t (1 /Y s ) f x ( s, /Y s ) ds (cid:12)(cid:12)(cid:12) Y = y i = E ˆ P h Z t X s f x ( s, X s ) ds (cid:12)(cid:12)(cid:12) X = 1 /y i . Thus, f x ( t, x ) = g ( t, /x ) = E ˆ P h Z t X s f x ( s, X s ) ds (cid:12)(cid:12)(cid:12) X = x i , which is the desired result.Since h and h ′ have linear growth at most, there is a positive constant c such that | Y t h ′ ( Y t )+ ηh ( Y t ) | Y η +1 t ≤ c Y η +3 t . From Eq.(B.8), we have | f x ( t, x ) | ≤ E ˜ P (cid:2) | Y t h ′ ( Y t ) + ηh ( Y t ) | Y η +1 t (cid:12)(cid:12) Y = x (cid:3) e − bt x − ≤ c E ˜ P [ Y η +3 t | Y = x ] e − bt x − ≤ c e − bt x − (B.14)for some positive constant c , which is independent of t and x. Here, we used Lemma B.1, whichgives that the expectation E ˜ P [ Y η +3 t | Y = x ] is uniformly bounded in ( t, x ) on [0 , ∞ ) × (0 , ∞ )since the constant A in the lemma satisfies 0 < A = η + 3 < ασ + 3 . Thus, | f b ( t, x ) | ≤ E ˆ P x h Z t X s | f x ( s, X s ) | ds i ≤ c Z t e − bs E ˆ P x [ X − s ] ds . By Lemma B.1, the expectation E ˆ P x [ X − s ] is bounded in s on [0 , ∞ ) since the expectation con-verges to a positive constant. Therefore, | f b ( t, x ) | ≤ c b Z t e − bs ds ≤ c (1 − e − bt ) ≤ c for some positive constant c . Since f ( T, ξ ) converges to a positive constant as T → ∞ , weconclude that (cid:12)(cid:12)(cid:12)(cid:12) T ∂ b p T p T + η (cid:12)(cid:12)(cid:12)(cid:12) = 1 T (cid:12)(cid:12)(cid:12)(cid:12) f b ( T, ξ ) f ( T, ξ ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ c T for some positive constant c . B.4 Sensitivity of a We study the large-time asymptotic behavior of the sensitivity of p T with respect to the param-eter a. In this section, assume that h is nonzero, nonnegative, continuously differentiable andthat h, h ′ have linear growth at most. From the decomposition in Eq.(B.1), it follows that ∂ a p T p T = f a ( T, ξ ) f ( T, ξ ) − ( ∂ a η ) ln ξ − T ∂ a λ . It can be easily shown that f is continuously differentiable in a by considering the densityfunction of X t or by using (Park, 2018, Theorem 4.8). We focus on the large-time behavior of f a ( T, ξ ) . Differentiate Eq.(B.4) in a, then − f at + 12 σ x f axx + ( b − αx ) xf ax − ( ∂ a α ) x f x = 0 , f a (0 , x ) = h ( x ) x η (ln x ) ∂ a η . t follows that f a ( t, x ) = E ˆ P x h − ( ∂ a α ) Z t X s f x ( s, X s ) ds + h ( X t ) X ηt (ln X t ) ∂ a η i . (B.15)However, to obtain this equality, the Feynman–Kac formula in Remark 3.3 cannot be applieddirectly because the volatility function σx / of X does not have linear growth. Instead, wedefine Y := 1 /X and g ( t, y ) := − f a ( t, /y ) . Then, Y is a CIR process dY t = ( α + σ − bY t ) dt − σ p Y t d ˆ B t , and Eq.(B.13) becomes − g t + 12 σ yg yy + ( α + σ − by ) g y + ( ∂ a α )(1 /y ) f x ( t, /y ) = 0 , g (0 , y ) = h (1 /y )(1 /y ) η (ln y ) ∂ a η . This PDE satisfies all the conditions in Remark 3.3. Note that the function ( ∂ a α )(1 /y ) f x ( t, /y )is bounded in y by Eq.(B.8) since Eq.(B.9) is bounded in ( t, x ) , moreover g (0 , y ) is boundedbelow since ∂ a η < y < y > h (1 /y )(1 /y ) η (ln y ) is bounded for large y. It follows that g ( t, y ) = E ˆ P h − ( ∂ a α ) Z t (1 /Y s ) f x ( s, /Y s ) ds − h (1 /Y t )(1 /Y t ) η (ln Y t ) ∂ a η (cid:12)(cid:12)(cid:12) Y = y i = E ˆ P h − ( ∂ a α ) Z t X s f x ( s, X s ) ds + h ( X t ) X ηt (ln X t ) ∂ a η (cid:12)(cid:12)(cid:12) X = 1 /y i . Thus, f a ( t, x ) = g ( t, /x ) = E ˆ P h − ( ∂ a α ) Z t X s f x ( s, X s ) ds + h ( X t ) X ηt (ln X t ) ∂ a η (cid:12)(cid:12)(cid:12) X = x i , which is the desired result.Using Lemma B.1 and Eq.(B.14), we have | f a ( t, x ) | ≤ c E ˆ P x h Z t e − bs ds i + c ≤ c b + c for some positive constants c and c . Since f ( T, ξ ) converges to a positive constant as T → ∞ , we conclude that (cid:12)(cid:12)(cid:12)(cid:12) T ∂ a p T p T + ∂ a λ (cid:12)(cid:12)(cid:12)(cid:12) = 1 T (cid:12)(cid:12)(cid:12)(cid:12) f a ( T, ξ ) f ( T, ξ ) − ( ∂ a η ) ln ξ (cid:12)(cid:12)(cid:12)(cid:12) ≤ c T for some positive constant c . Furthermore, direct calculation gives ∂ a λ = − bσ p ( a + σ / + 2 qσ − ( a + σ / p ( a + σ / + 2 qσ . B.5 Sensitivity of σ We study the large-time asymptotic behavior of the sensitivity of p T with respect to the param-eter σ. In this section, assume that h is nonzero, nonnegative, twice continuously differentiableand that h, h ′ have linear growth at most and h ′′ is bounded. From the decomposition inEq.(B.1), it follows that ∂ σ p T p T = f σ ( T, ξ ) f ( T, ξ ) − ( ∂ σ η ) ln ξ − T ∂ σ λ . t can be easily shown that f is continuously differentiable in σ by considering the densityfunction of X t or by using (Park, 2018, Theorem 4.13). We focus on the large-time behavior of f σ ( T, ξ ) . Differentiate Eq.(B.4) in σ, then − f σt + 12 σ x f σxx + ( b − αx ) xf σx + σx f xx − ( ∂ σ α ) x f x = 0 , f (0 , x ) = h ( x ) x η (ln x ) ∂ σ η . It follows that f σ ( t, x ) = E ˆ P x h σ Z t X s f xx ( s, X s ) ds − ( ∂ σ α ) Z t X s f x ( s, X s ) ds + h ( X t ) X ηt (ln X t ) ∂ σ η i = E ˆ P x h σ Z t X s f xx ( s, X s ) ds i + E ˆ P x h − ( ∂ σ α ) Z t X s f x ( s, X s ) ds + h ( X t ) X ηt (ln X t ) ∂ σ η i . We claim that the two expectations on the right-hand side are bounded in t on [0 , ∞ ) . The secondexpectation is bounded in t on [0 , ∞ ) by the same method used in the analysis of Eq.(B.15). Toestimate the first expectation, observe that f xx ( t, x ) = g x ( t, x ) e − bt x − − g ( t, x ) e − bt x − , which is presented in Eq.(B.11). It follows that E ˆ P x h Z t X s f xx ( s, X s ) ds i = E ˆ P x h Z t X s g x ( s, X s ) e − bs ds i − E ˆ P x h Z t g ( s, X s ) e − bs ds i . By the same method used in Eq.(B.14), there is a positive constant c , which is independent of t and x, such that | g ( t, x ) | ≤ E ˜ P (cid:2) | Y t h ′ ( Y t ) + ηh ( Y t ) | Y η +1 t (cid:12)(cid:12) Y = x (cid:3) ≤ c . Thus, E ˆ P x [ R t g ( s, X s ) e − bs ds ] is bounded in t on [0 , ∞ ) . From Eq.(B.12), | g x ( t, x ) | ≤ E ¯ P (cid:2) | h ′′ ( Z t ) Z t + 2( η + 1) Z t h ′ ( Z t ) + η ( η + 1) h ( Z t ) | Z η +2 t (cid:12)(cid:12) Z = x (cid:3) e − bt x − ≤ c E ¯ P [ Z η +4 t (cid:12)(cid:12) Z = x ] e − bt x − for some positive constant c . By Lemma B.1, the expectation E ¯ P [ Z η +4 t (cid:12)(cid:12) Z = x ] is uniformlybounded in ( t, x ) on the domain [0 , ∞ ) × (0 , ∞ ) , which implies that | g x ( t, x ) | ≤ c e − bt x − for some positive constant c . Thus, E ˆ P x h Z t X s | g x ( s, X s ) | e − bs ds i ≤ c E ˆ P x h Z t X − s e − bs ds i = c Z t E ˆ P x [ X − s ] e − bs ds . By Lemma B.1, the expectation E ˆ P x [ X − s ] is bounded in s on [0 , ∞ ) since the expectation con-verges to a positive constant, which gives the desired result. In conclusion, (cid:12)(cid:12)(cid:12)(cid:12) T ∂ σ p T p T + ∂ σ λ (cid:12)(cid:12)(cid:12)(cid:12) = 1 T (cid:12)(cid:12)(cid:12)(cid:12) f σ ( T, ξ ) f ( T, ξ ) − ( ∂ σ η ) ln ξ (cid:12)(cid:12)(cid:12)(cid:12) ≤ c T for some positive constant c . Furthermore, direct calculation gives ∂ σ λ = b ( a + σ / q − p ( a + σ / + 2 qσ ) σ p ( a + σ / + 2 qσ − bσ ( p ( a + σ / + 2 qσ − a − σ / . eferences Abramowitz, M. and Stegun, I. A. (1965).
Handbook of mathematical functions: with formulas,graphs, and mathematical tables , volume 55. Courier Corporation.Delbaen, F. and Schachermayer, W. (1994). A general version of the fundamental theorem ofasset pricing.
Mathematische annalen , 300(1):463–520.Dereich, S., Neuenkirch, A., and Szpruch, L. (2011). An Euler-type method for the strongapproximation of the Cox–Ingersoll–Ross process.
Proceedings of the Royal Society A: Math-ematical, Physical and Engineering Sciences , 468(2140):1105–1115.Fleming, W. H. and McEneaney, W. M. (1995). Risk-sensitive control on an infinite time horizon.
SIAM Journal on Control and Optimization , 33(6):1881–1915.Fourni´e, E., Lasry, J.-M., Lebuchoux, J., Lions, P.-L., and Touzi, N. (1999). Applications ofMalliavin calculus to Monte Carlo methods in finance.
Finance and Stochastics , 3(4):391–412.Gobet, E. and Munos, R. (2005). Sensitivity analysis using Itˆo-Malliavin calculus and martin-gales, and application to stochastic optimal control.
SIAM Journal on Control and Optimiza-tion , 43(5):1676–1713.Hansen, L. P. (2012). Dynamic valuation decomposition within stochastic economies.
Econo-metrica , 80(3):911–967.Hansen, L. P. and Scheinkman, J. (2009). Long-term risk: An operator approach.
Econometrica ,77(1):177–234.Hansen, L. P. and Scheinkman, J. (2012). Pricing growth-rate risk.
Finance and Stochastics ,16(1):1–15.Hurd, T. R. and Kuznetsov, A. (2008). Explicit formulas for Laplace transforms of stochasticintegrals.
Markov Processes and Related Fields , 14(2):277–290.Jeanblanc, M., Yor, M., and Chesney, M. (2009).
Mathematical methods for financial markets .Springer Science & Business Media.Karatzas, I. and Shreve, S. (1991).
Brownian Motion and Stochastic Calculus , volume 113.Springer Science & Business Media.Kramkov, D. and Schachermayer, W. (1999). The asymptotic elasticity of utility functions andoptimal investment in incomplete markets.
The Annals of Applied Probability , 9(3):904–950.Kramkov, D. and Sˆırbu, M. (2006). Sensitivity analysis of utility-based prices and risk-tolerancewealth processes.
The Annals of Applied Probability , 16(4):2140–2194.Liu, R. and Muhle-Karbe, J. (2013). Portfolio choice with stochastic investment opportunities:a user’s guide. arXiv preprint arXiv:1311.1715 .Mostovyi, O. (2018). Asymptotic analysis of the expected utility maximization problem withrespect to perturbations of the num´eraire. arXiv preprint arXiv:1805.11427 .Mostovyi, O. and Sˆırbu, M. (2017). Sensitivity analysis of the utility maximization problemwith respect to model perturbations. arXiv preprint arXiv:1705.08291 . ark, H. (2018). Sensitivity analysis of long-term cash flows. Finance and Stochastics , 22(4):773–825.Park, H. and Sturm, S. (2019). A sensitivity analysis of the long-term expected utility of optimalportfolios.
Available at SSRN 3401532 .Pinsky, R. G. (1995).
Positive Harmonic Functions and Diffusion , volume 45. CambridgeUniversity Press.Qin, L. and Linetsky, V. (2016). Positive eigenfunctions of Markovian pricing operators:Hansen-Scheinkman factorization, Ross recovery, and long-term pricing.
Operations Research ,64(1):99–117.Robertson, S. and Xing, H. (2015). Large time behavior of solutions to semilinear equations withquadratic growth in the gradient.
SIAM Journal on Control and Optimization , 53(1):185–212., 53(1):185–212.