Non-linear filtering and optimal investment under partial information for stochastic volatility models
aa r X i v : . [ q -f i n . P M ] J u l Non-linear filtering and optimal investment under partialinformation for stochastic volatility models
Dalia Ibrahim; Frédéric Abergel ∗ ;July 28, 2015 Abstract
This paper studies the question of filtering and maximizing terminal wealth from ex-pected utility in a partially information stochastic volatility models. The special features isthat the only information available to the investor is the one generated by the asset prices,and the unobservable processes will be modeled by a stochastic differential equations. Usingthe change of measure techniques, the partial observation context can be transformed intoa full information context such that coefficients depend only on past history of observedprices (filters processes). Adapting the stochastic non-linear filtering, we show that undersome assumptions on the model coefficients, the estimation of the filters depend on a priorimodels for the trend and the stochastic volatility. Moreover, these filters satisfy a stochasticpartial differential equations named "Kushner-Stratonovich equations". Using the martin-gale duality approach in this partially observed incomplete model, we can characterize thevalue function and the optimal portfolio. The main result here is that the dual value func-tion associated to the martingale approach can be expressed, via the dynamic programmingapproach, in terms of the solution to a semilinear partial differential equation. We illus-trate our results with some examples of stochastic volatility models popular in the financialliterature.
Keywords 0.1.
Partial information, stochastic volatility, utility maximization, martingale du-ality method, non-linear filtering, Kushner-Stratonovich equations, semilinear partial differentialequation.
The basic problem of mathematical finance is the problem of an economic agent who invests in afinancial market so as to maximize the expected utility of his terminal wealth. In the frameworkof continuous time model, the utility maximization problem has been studied for the first timeby Merton (1971) in a Black-Scholes environment (full information) via the Hamilton-Jaccobi-Bellman equation and dynamic programming. As in financial market models, we do not havein general a complete knowledge of all the parameters, which may be driven by unobservedrandom factors. So, we are in the situation of the utility maximization problem with partialobservation, which has been studied extensively in the literature by Detemple [ ? ], Dothan andFeldman [ ? ], Lakner [ ? ], [ ? ], etc. There are many generalizations of Merton’s setting. Thenatural generalizations was to model the volatility by a stochastic process.In this paper, we consider a financial market where the price process of risky asset follows astochastic volatility model and we require that investors observe just the stock price. So we are ∗ Ecole Centrale Paris, Laboratoire de Mathématiques Appliqués aux Systèmes, Grande Voie des Vignes, 92290Châtenay Malabry, France
1n the framework of partially observed incomplete market, where our aim is to solve the utilitymaximization problem in this context.In order to solve this problem with partial observation, the common way is to use the stochas-tic non-linear filtering and change of measure techniques, so as the partial observation contextcan be transformed into a full information context. Then it is possible to solve this problem eitherwith the martingale approach or via dynamic programming approach. Models with incompleteinformation have been investigated by Dothan and Feldman [ ? ] using dynamic programmingmethods in a linear Gaussian filtering, Lakner [ ? ], [ ? ] has solved the partial optimization prob-lem via martingale approach and worked out the special case of the linear Gaussian filtering.Pham and Quenez [ ? ] treated the case of partial information stochastic volatility model wherethey have combined stochastic filtering techniques and a martingale duality approach to char-acterize the value function and the optimal portfolio of the utility maximization problem. Theyhave studied two cases: the case where the risks of the model are assumed to be independentGaussian processes and the Bayesian case studied by Karatzas-Zhao [ ? ].In this paper, we are in the same framework studied by Pham and Quenez [ ? ], but here weassume that the unobservable processes are modeled by a stochastic differential equations. Moreprecisely, the unobservable drift of the stock and that of the stochastic volatility are modeledby stochastic differential equations. The main result in this case, is that the filters estimate ofthe risks depend on a priori models for the trend and the stochastic volatility. There are tworeasons for this result: Firstly, we need to choose the models of the trend and the stochasticvolatility such that the risks dynamics can be described only in terms of them. Secondly, we needto choose these models such that the coefficients of the risks dynamics satisfy some regularityassumptions, like globally Lipshitz conditions and some finite order moment will be imposed. Weshow that the filters estimate of the risks satisfy a stochastic partial differential equations named"Kushner-Stratonovich equations". But these equations are valued in infinite dimensional spaceand cannot be solved explicitly, so numerical approximitions can be used to resolve them. Also,we study the case of finite dimensional filters like Kalman-Bucy filter. We illustrate our resultswith several popular examples of stochastic volatility models.After replacing the original partial information problem by a full information one whichdepends only on the past history of observed prices, it is then possible to use the classicaltheory for stochastic control problem. Here we will be interested by the martingale approach tosolve our utility optimization problem. As the reduced market in incomplete, we complementthe martingale approach by using the theory of stochastic control to solve the related dualoptimization problem. In [ ? ], they have also used the martingale approach, but they havestudied the case where the dual optimizer vanishes. The main result in this paper is that thesolution of the related dual problem can be expressed in terms of the solution to a semilinearpartial differential equation which depends also on the filters and the stochastic volatility.The paper is organized as follows: In section , we describe the model and formulate theoptmization problem. In section , we use the non-linear filtering techniques and the change ofmeasure techniques in order to transform the partial observation context into a full informationcontext such that coefficients depend only on past history of observed prices (filters processes).In section , we show that the filters estimations depend on a priori models for the trend andthe stochastic volatility. We illustrate our results with examples of stochastic volatility modelspopular in the financial literature. Finally, in section , we use the martingale duality approachfor the utility maximization problem. We show that the dual value function and the dualoptimizer can be expressed in terms of the solution to a semilinear partial differential equation.By consequence, the primal vale function and the optimal portfolio depend also on this solution.The special cases of power and logarithmic utility functions are studied and we illustrate ourresults by an examples of stochastic volatility models for which we can give a closed form to the2emilinear equation. Let (Ω , F , P ) be a complete probability space equipped with a filtration F = {F t , ≤ t ≤ T } satisfying the usual conditions, where T > is a fixed time horizon. The financial marketconsists of one risky asset and a bank account (bound). The price of the bound is assumed forsimplicity to be over the entire continuous time-horizon [0 , T ] and the risky asset has dynamics: dS t S t = µ t dt + g ( V t ) dW t , (2.1) dV t = f ( β t , V t ) dt + k ( V t )( ρdW t + p − ρ dW t ) , (2.2) dµ t = ζ ( µ t ) dt + ϑ ( µ t ) dW t . (2.3)The processes W and W are two independents Brownian motions defined on (Ω , F , P ) and − ≤ ρ ≤ is the correlation coefficient. W is a standard Brownian motion independent of W and W . The drift µ = { µ t , ≤ t ≤ T } is not observable and follows a Gaussian process.The process β t can be taken as a function in terms of µ t or another unobservable process, whichalso has a stochastic differential equation.We assume that the functions g , f , k , ζ and ϑ ensure existence and uniqueness for solutions tothe above stochastic differential equations. A Lipschitz conditions are sufficient, but we do notimpose these on the parameters at this stage, as we do not wish to exclude some well-knownstochastic volatility models from the outset. Also, we can assume that the drift µ t can bereplaced by µ t g ( V t ) , that is we have a factor model.Moreover, we assume that g ( x ) , k ( x ) > and the solution of (2.2) does not explode, that is, thesolution does not touch or ∞ in finite time. The last condition can be verified form Feller’stest for explosions given in [ ? , p.348].In the sequel, we denote by F S = { F St , ≤ t ≤ T } (resp. F V = { F Vt , ≤ t ≤ T } ) the filtrationgenerated by the price process S (resp. by the stochastic volatility V ). Also we denote by G = {G t , ≤ t ≤ T } the natural P -augmentation of the market filtration generated by the priceprocess S . Let π t be the fraction of the wealth that the trader decides to invest in the risky asset at time t ,and − π t is the fraction of wealth invested in the bound. We assume that the trading strategyis self-financing, then the wealth process corresponding to a portfolio π is defined by R π = x and satisfies the following SDE : dR πt = R πt (cid:0) π t µ t dt + π t g ( V t ) dW t (cid:1) . A function U : R → R is called a utility function if it is strictly increasing, strictly concave ofclass C . We assume that the investor wants to maximize the expected utility of his terminalwealth. The optimization problem thus reads as J ( x ) = sup π ∈A E [ U ( R πT )] , x > , (2.4)3here A denotes the set of the admissible controls ( π t , ≤ t ≤ T ) which are F S -adapted, andsatisfies the integrability condition: Z Tt g ( V s ) π s ds < ∞ P − a.s. (2.5)We are in a context when an investor wants to maximize the expected utility from terminalwealth, where the only information available to the investor is the one generated by the assetprices, therefore leading to a utility maximization problem in partially observed incompletemodel. In order to solve it, we aim to reduce it to a maximization problem with full information.For that, it becomes important to exploit all the information coming from the market itself inorder to continuously update the knowledge of the not fully known quantities and this is wherestochastic filtering becomes useful. Let us consider the following processes: ˜ µ t := µ t g ( V t ) , (3.1) ˜ β t := (cid:16)p − ρ k ( V t ) (cid:17) − ( f ( β t , V t ) − ρk ( V t )˜ µ t ) , (3.2)we assume that they verify the integrability condition: Z T | ˜ µ t | + | ˜ β t | dt < ∞ a . s . Here ˜ µ t and ˜ β t are the unobservable processes that account for the market price of risk. The firstis related to the asset’s Brownian component. The second to the stochastic volatility’s Brownianmotion.Also we introduce the following process: L t = 1 − Z t L s h ˜ µ s dW s + ˜ β s dW s i . (3.3)We shall make the usual standing assumption of filtering theory. Assumption 1.
The process L is a martingale, that is, E [ L T ] = 1 . Under this assumption, we can now define a new probability measure ˜ P equivalent to P on (Ω , F ) characterized by: d ˜ P d P |F t = L t , ≤ t ≤ T. (3.4)Then Girsanov’s transformation ensures that ˜ W t = W t + Z t ˜ µ s ds is a (˜ P , F ) -Brownian motion , (3.5) ˜ W t = W t + Z t ˜ β s ds is a (˜ P , F ) -Brownian motion . (3.6)(3.7)4lso, we have that (˜ µ t , ˜ β t ) is independent of the Brownian motion (cid:16) ˜ W t , ˜ W t (cid:17) .Therefore, the dynamics of ( S, V ) under ˜ P become: dS t S t = g ( V t ) d ˜ W t , (3.8) dV t = ρ k ( V t ) d ˜ W t + p − ρ k ( V t ) d ˜ W t . (3.9)We now state a lemma which will highly relevant in the following. The proof of this lemma issimilar to lemma . in Pham and Quenez [ ? ]. Lemma 3.1.
Under assumption 1, the filtration G is the augmented filtration of ( ˜ W , ˜ W ) .Proof. The sketch of the proof is summarized by two steps:Firstly, we show that the filtration G is equal to the enlarged progressive filtration F S ∨ F V . Thefirst inclusion is obvious and the other inclusion F S ∨ F V ⊂ G is deduced from the fact that V t can be estimated from the quadratic variation of log ( S t ) . Secondly, from (3.8), (3.9) and thefact that g ( x ) , k ( x ) > , we have that F ˜ W W F ˜ W the filtration generated by ( ˜ W , ˜ W ) We now make the following assumption on the risk processes (cid:16) ˜ µ, ˜ β (cid:17) . ∀ t ∈ [0 , T ] , E | ˜ µ t | + E | ˜ β t | < ∞ (3.10)Under this assumption, we can introduce the conditional law of (cid:16) ˜ µ, ˜ β (cid:17) : µ t := E [˜ µ t |G t ] , (3.11) β t := E [ ˜ β t |G t ] . (3.12)Let us denote by H the (˜ P , F ) martingale defined as H t = 1 L t . Now, we aim to construct therestriction of P equivalent to ˜ P on (Ω , G ) . First, let us consider the conditional version of Baye’sformula: for any P integrable random variable X ( X ∈ L ( P ) ), we have: E [ X |G t ] = ˜ E [ XH t |G t ]˜ E [ H t |G t ] . (3.13)Then by taking X = L t , we get: ˜ L t := E [ L t |G t ] = 1˜ E [ H t |G t ] . (3.14)Therefore, from (3.4) (3.14), we have the following restriction to G : d ˜ P d P |G t = ˜ L t . Finally, from Bain and Crisan (proposition . ) and Pardoux (proposition . . ), we have thefollowing result: 5 roposition 3.2. The following processes W and W are independent ( P , G ) -Brownian mo-tions. W t = W t + Z t (˜ µ s − µ s ) ds := ˜ W t − Z t µ s ds,W t = W t + Z t (cid:16) ˜ β s − β s (cid:17) ds := ˜ W t − Z t β s ds. These processes are called the innovation processes in filtering theory. They include the distancesbetween the true values of ˜ µ and ˜ β and their estimates:Then, by means of the innovation processes, we can describe the dynamics of ( S, V, R ) within aframework of full observation model: ( Q ) = dS t S t = g ( V t ) µ t dt + g ( V t ) dW t ,dV t = (cid:16) ρ k ( V t ) µ t + p − ρ k ( V t ) β t (cid:17) dt + ρk ( V t ) dW t + p − ρ k ( V t ) dW t ,dR πt = R πt π t (cid:16) g ( V t ) µ t dt + g ( V t ) dW t (cid:17) . We have showed that conditioning arguments can be used to replace the initial partial infor-mation problem by a full information problem one which depends only on the past history ofobserved prices. But the reduction procedure involves the filters estimate µ t and β t .Our filtering problem can be summarized as follows: From lemma 3.1, we have G = F ˜ W ∨ F ˜ W . Then the vector ( ˜ W , ˜ W ) corresponds to the observation process. On the other hand,our signal process is given by ( ˜ µ t , ˜ β t ) . So the filtering problem is to characterize the conditionaldistribution of ( ˜ µ t , ˜ β t ) , given the observation data G = F ˜ W W ˜ W .We show in this section how the filters estimate depend on the models of the drift andthe stochastic volatility. Using the non-linear filtering theory (presenting in appendix), wecan deduce that the filters estimate satisfy some stochastic partial differential equations, called"Kushner-Stratonovich equations". Generally these equations are infinite-dimensional and thusvery hard to solve them explicitly. So, in order to simplify the situation and in order to obtaina closed form for the optimal portfolio, we will be interested by some cases of models, when wecan deduce a finite dimensional filters. Let us assume that the processes ˜ µ t and ˜ β t are solutions of the following stochastic differentialequations: d (cid:18) ˜ µ t ˜ β t (cid:19) = (cid:18) aa (cid:19) dt + (cid:18) g g g g (cid:19) d (cid:18) W t W t (cid:19) + (cid:18) b b b b (cid:19) d (cid:18) W t W t (cid:19) (4.1)where we denote for simplification the functions a := a (˜ µ t , ˜ β t ) , a := a (˜ µ t , ˜ β t ) , ....... b = b (˜ µ t , ˜ β t ) , and the Brownian motion ( W t , W t ) is independent of ( W t , W t ) .6n the other hand, the dynamics of the observation process ( ˜ W , ˜ W ) is given by: d (cid:18) ˜ W t ˜ W t (cid:19) = d (cid:18) W t W t (cid:19) + (cid:18) ˜ µ t ˜ β t (cid:19) dt (4.2) Remark 4.1.
To avoid confusion in the sequel, we have: G = F ˜ W W ˜ W = F Y . Notations 1.
Let us denote by: X t = (cid:18) ˜ µ t ˜ β t (cid:19) , Y t = (cid:18) ˜ W t ˜ W t (cid:19) , A = (cid:18) aa (cid:19) , G = (cid:18) g g g g (cid:19) , B = (cid:18) b b b b (cid:19) M t = (cid:18) W t W t (cid:19) W t = (cid:18) W t W t (cid:19) , h = (cid:18) h h (cid:19) K = 12 ( BB T + GG T ) . (4.3) where for x = ( m, b ) , h ( x ) = m , h ( x ) = b and T denotes the the transposition operator. With these notations, the signal-observation processes ( X t , Y t ) satisfy (A.1) and (A.2): dX t = A ( X t ) dt + G ( X t ) dM t + B ( X t ) dW t (4.4) dY t = dW t + h ( X t ) dt (4.5) µ t and β t Let us now make some assumptions which will be useful to show our results.
Assumptions • i ) The functions
A, G and B are globally Lipschitz. • ii ) X has finite second moment. • iii ) X has finite third moment. Lemma 4.2.
Let ( X, Y ) be the solution of (4.4) and (4.5) and assume that h has linear growthcondition. If assumptions i ) and ii ) are satisfied, then (A.4) is satisfied. Moreover, if assumption iii ) is satisfied, then (A.6) is satisfied.Proof. The proof is given in [ ? ](see, lemma . . and lemma . . ).The following results show that we need to introduce an a priori models for the trend andthe stochastic volatility in order to describe the dynamics of (˜ µ t , ˜ β t ) as in (4.1), and thereforewe can deduce from proposition A.2 the dynamics of the filters estimate ( µ t , β t ) and thereforededuce that of ( µ t , β t ) . More precisely, we show that these estimates depend essentially on themodel of the volatility V t . We need to choose the dynamics of V t such that the following twosteps will be verified. • First step: Describe the dynamics of (˜ µ t , ˜ β t ) as in (4.1)We show that this description depend essentially on the model of V t . In fact, if we applyItô’s formula on ˜ µ t and ˜ β t in order to describe their dynamics, we have that V t still appear,for that we need to describe V t only in terms of ˜ µ t and ˜ β t in order to disappear it fromtheir dynamics. This can be done from the definition of the ˜ β t but taking in account thechoice of the variable β t or more precisely the choice of f ( β t , V t ) . We will clarify this withan examples in paragraph . . . 7 Second step: Verification of some regularity assumptionsOnce we describe the dynamics of (˜ µ t , ˜ β t ) as in (4.1), we must check in more that thecoefficients of the dynamics verify some regularity assumptions, in order to use the aboveresults of nonlinear filtering theory.We present now our result concerning the filtering problem: Proposition 4.3.
We assume that there exists a function
Υ : R → R such that V t = Υ(˜ µ t , ˜ β t ) .If with this function, the dynamics of X t = (˜ µ t , ˜ β t ) can be described as in (4.4) and assump-tions i ) , ii ) and iii ) hold, then the conditional distribution α t : E [ φ ( X t | F Yt )] satisfy the followingKushner-Stratonovich equation: dα t ( φ ) = α t ( Aφ ) dt + (cid:2) α t (cid:0)(cid:0) h + B (cid:1) φ (cid:1) − α t ( h ) α t ( φ ) (cid:3) dW t + (cid:2) α t (cid:0)(cid:0) h + B (cid:1) φ (cid:1) − α t ( h ) α t ( φ ) (cid:3) dW t . (4.6) for any φ ∈ B ( R ) (the space of bounded measurable functions R → R ). The operators B and B are given in (A.9). Moreover the dynamics of ( µ t , β t ) satisfy the following stochasticdifferential equations: dµ t = α t ( a ) dt + [ α t (cid:0) h φ + b (cid:1) − α t ( h ) α t ( φ )] dW t + [ α t (cid:0) h φ + b (cid:1) − α t ( h ) α t ( φ )] dW t ,dβ t = α t ( a ) dt + [ α t (cid:0) h φ + b (cid:1) − α t ( h ) α t ( φ )] dW t + [ α t (cid:0) h φ + b (cid:1) − α t ( h ) α t ( φ )] dW t . Proof.
From the definition of ˜ µ t and ˜ β t and depending on the models of µ t and β t , we havefrom Itô’s formula that V t still appear in the dynamics of ˜ µ t and ˜ β t . As V t = Υ(˜ µ t , ˜ β t ) , then wecan describe the dynamics of the signal process X t = (˜ µ, ˜ β t ) as in (A.1). On the other hand,from the definition of the observation process given by (4.5), we have that the sensor function h = ( h , h ) has a linear growth condition. Thus, as assumptions i ) , ii ) and iii ) are verified,then we can deduce from lemma 4.2, that the conditions (A.4) and (A.6) are proved. Thereforethe dynamics of α t given in (4.6) is deduced from proposition A.2.It remains to deduce the dynamics of ( µ t , β t ) .Let us consider the functions φ and φ as follows:for x = ( m, b ) , φ ( x ) = m and φ ( x ) = b. Then the filters µ t (resp. β t ) can be deduce from (4.6) by replacing φ by φ (resp. φ ). The problemhere is that the Kushner-Stratonovich equation (4.6) holds for any bounded Borel measurable φ . But as φ (resp. φ ) not bounded, we proceed by truncating of φ (resp. φ ) at a fixed levelwhich we let tend to infinity. For this, let us introduce the functions ( ψ k ) k> defined as ψ k ( x ) = ψ ( x/k ) , x in R , where ψ ( x ) = if | x | ≤ | x | − | x | − ) if < | x | < if | x | ≥ . Then by using the following relations given in: lim k →∞ φ ψ k ( x ) = φ ( x ) , | φ ( x ) ψ k ( x ) | ≤ | φ ( x ) | , lim k →∞ A s ( φ ψ k )( x ) = A s φ ( x ) . Then by replacing in equation (4.6) φ by φ ψ k and from dominated convergence theorem, wemay pass to the limit as k → ∞ and then we deduce that µ t := α t ( φ ) (resp. β t := α t ( φ ) )satisfy the dynamics given above. 8 .1.2 Existence and uniqueness of the solution to equation (4.6) We now take sufficient assumption on the coefficients of the signal-observation system in orderto show that equation (4.6) has a unique solution, see Bain and Crisan [ ? , chap.4]. We definein the following the space within which we prove the uniqueness.Let us define the space of measure-valued stochastic processes within which we prove uniquenessof the solution to equation (4.6). This space has to be chosen so that it contains only measureswith respect to which the integral of any function with linear growth is finite. The reason ofthis choice is that we want to allow to the coefficients of the signal and observation processes tobe unbounded.Let ψ : R → R be the function ψ ( x ) = 1 + || x || , for any x ∈ R and define C l ( R ) to bethe space of continuous functions φ such that φ/ψ ∈ C b ( R ) (the space of bounded continuousfunctions).Let us denote by M l ( R ) the space of finite measure M such that M ( ψ ) < ∞ . In particular, thisimplies that µ ( φ ) < ∞ for all φ ∈ C l ( R ) . Moreover, we endow M l ( R ) wit the correspondingweak topology: A sequence ( µ n ) of measures in M l ( R ) converges to µ ∈ M l ( R ) if and only if lim n →∞ µ n ( φ ) = µ ( φ ) , for all φ ∈ C l ( R ) . Definition 4.4. • The Class U is the space of all Y t -adapted M l ( R ) -valued stochastic pro-cess ( µ ) t > with càdlàg paths such that, for all t > , we have ˜ E (cid:20)Z t ( µ s ( ψ )) ds (cid:21) < ∞ . • The Class ˜ U is the space of all Y t -adapted M l ( R ) -valued stochastic process ( µ ) t > withcàdlàg paths such that the process m µ µ belongs to the class U , where the process m µ isdefined as: m µt = exp( Z t µ s ( h T ) dY s − Z t µ s ( h T ) µ s ( h ) ds ) Now we state the uniqueness result of the solution to equation (4.6), see theorem . inBain and Crisan [ ? , chap.4] Proposition 4.5.
Assuming that the functions A , K and h defined in (4.3) have twice contin-uously differentiable components and all their derivatives of first and second order are bounded.Then equation (4.6) has a unique solution in the class ˜ U . Remark 4.6.
The equations satisfied by the filters are infinite-dimensional and cannot be solvedexplicitly. These filters have to be solved numerically, but in concrete application, the filtercould thus never be implemented exactly, so in order to avoid this difficulty, some approximationschemes have been proposed. For example, the extended Kalman filter, which is based uponlinearization of the state equation around the current estimate, see e.g Pardoux [ ? ]. This methodis not mathematically justified, but it is widely used in practice. The partial differential equationsmethod which based on the fact that the density of the unnormalised conditional distribution ofthe signal is the solution of a partial differential equation, see e.g Bensoussan[ ? ] and Pardoux [ ? ].Also, we can use the approximation scheme used by Gobet el al [ ? ] which consist in discretizing theZakai equation, which is linear, and then deduce the approximation of the conditional distribution α t from Kllianpur-Striebel formula (A.5). .1.3 Application In this section, we will present two types of models: a models for which we cannot apply ourresult in proposition 4.3 in order to deduce the filters estimate and a models where proposition4.3 can be applied.Let us consider the following dS t S t = µ t dt + e V t dW t (4.7) dV t = λ V ( θ − V t ) dt + σ V ρdW t + σ V p − ρ dW t (4.8) dµ t = λ µ ( θ µ − µ t ) dt + σ µ dW t , µ N ( m , σ ) , (4.9)Here the risks of the models are given by: ˜ µ t = µ t e V t ˜ β t = λ V ( θ − V t ) σ V p − ρ − ρ p − ρ ˜ µ t Applying Itô’s formula on ˜ µ t and β t , we have the following dynamics: ˜ µ t = ˜ µ + Z t λ µ θ µ e − V s ds + ˜ µ s (cid:18) σ V − λ µ − (cid:20) λ V ( θ − V s ) + 12 σ V (cid:21)(cid:19) ds + Z t σ V e − V s dW s − Z t ρσ V ˜ µ s dW s − Z t p − ρ σ V ˜ µ s dW s . ˜ β t = − Z t λ V ( θ − V s ) σ V p − ρ ds − Z t λ V ρ p − ρ dW s − Z t λ V dW s − ρ p − ρ d ˜ µ s . On the other hand, from the definition of ˜ β t , we can express V t in terms of ˜ µ t and ˜ β t as follows: V t = − σ V p − ρ λ V ˜ β t − σ V ρλ V ˜ µ t + θ. (4.10)If we replace V t in the above dynamics, we can deduce that (˜ µ t , ˜ β t ) can be described as in (A.1),where: a ( m, b ) = λ µ θ µ exp σ V p − ρ λ V b + σ V ρλ V m − θ ! + (cid:18) σ V − λ µ − σ V p − ρ b − σ V ρm (cid:19) m ; b ( m, b ) = − ρσ V m ; b ( m, b ) = − σ V m p − ρ ; g ( m, b ) = σ V exp σ V p − ρ λ V b + σ V ρλ V m − θ ! . and a ( m, b ) = − λ V b − λ V ρ p − ρ m − ρ p − ρ a ( m, b ); b ( m, b ) = − λ V ρ p − ρ − ρ σ V p − ρ mb ( m, b ) = − λ V + ρσ V m ; g ( m, b ) = − ρ p − ρ exp σ V p − ρ λ V b + σ V ρλ V m − θ ! ; g = g = 0 . With (4.10), the dynamics of (˜ µ t , ˜ β t ) is described as in (A.1) but assumption i ) about theglobally Lipschitz conditions is not satisfied, then proposition 4.3 can’t be applied.10 emark 4.7. Notice that here β t is a constant function. Also we can choose for example β t = µ t which in this case we can still describe V t only in terms of ˜ µ t and ˜ β t . But if we take β t is anotherprocess, in this case it is not clear that V t can be described only in terms of ˜ µ t and ˜ β t . Let us now consider another example: Heston model dS t S t = µ t dt + p V t dW t ,dV t = λ V ( θ − V t ) dt + σ V p V t (cid:16) ρdW t + p − ρ dW t (cid:17) ,dµ t = λ µ ( θ µ − µ t ) dt + σ µ dW t , µ N ( m , σ ) , Here the risks are given by ˜ µ t = µ t √ V t and ˜ β t = λ V ( θ − V t ) σ V √ V t p − ρ − ρ p − ρ ˜ µ t . Also here weare in the above situation that is we can describe the dynamics of ˜ µ t and ˜ β t as in (4.3), butassumption i ) is not satisfied.Now we give some examples with which proposition (4.3) can be applied and therefore wecan deduce the filters estimate. we will be interested by the stochastic factor Garch model andthe stochastic factor Log Ornstein-Uhlenbeck model.Stochastic factor Garch model:Let us consider the following Garch-model: dS t S t = p V t (cid:0) µ t dt + dW t (cid:1) ,dV t = β t ( θ − V t ) dt + σ V V t (cid:16) ρdW t + p − ρ dW t (cid:17) ,dµ t = λ µ ( θ µ − µ t ) dt + σ µ dW t , µ N ( m , σ ) ,dβ t = λ β β t dt + σ β dW t β N ( m , σ ) . where W and W are independent and independent from W and W where µ and β followrespectively a normal distribution of mean m (resp. m ) and variance σ (resp. σ ).Here the risk of the model are given by: ˜ µ t = µ t and ˜ β t = β t ( θ − V t ) p − ρ V t − ρ p − ρ ˜ µ t . In order to compute the filters estimate in this case of models, we will be interested by usingproposition 4.3. For that, we need to take θ = 0 . Because, if we apply Itô’s formula on ˜ µ t and ˜ β t in the case where θ = 0 , we obtain a dynamics with coefficients are not Lipschitz, that is,assumption i ) is not verify and therefore proposition 4.3 can’t be applied. For that we will take θ = 0 . Let θ = 0 , then from Itô’s formula, we have: d (cid:18) ˜ µ t ˜ β t (cid:19) = A (cid:18) ˜ µ t ˜ β t (cid:19) dt + G (cid:18) ˜ µ t ˜ β t (cid:19) dM t . where the functions A, G and B are given as follows: A (cid:18) mb (cid:19) = λ µ ( θ µ − m ) λ β b + ρ ( λ β + λ µ ) ρ m − ρλ µ θ µ ρ , G (cid:18) mb (cid:19) = − ρσ µ ρ σ β ( b + ρρ m ) . ρ = p − ρ and the function B is null, so we are in the case where the signal process X t := (˜ µ t , ˜ β t ) and the observation processes Y t := ( ˜ W t , ˜ W t ) are independent. This implies thatthe operator B and B will disappear in the Zakai and Kushner-Stratonovich equations. As forthis model, the assumptions of proposition 4.3 are satisfied, then the conditional distribution α t is given for any φ by: dα t ( φ ) = α t ( A φ ) dt + (cid:2) α t (cid:0) h φ (cid:1) − α t ( h ) α t ( φ ) (cid:3) dW t + (cid:2) α t (cid:0) h φ (cid:1) − α t ( h ) α t ( φ ) (cid:3) dW t . Here the operator A is given by (A.7), where K = 12 GG T .Therefore, the dynamics of the filter estimate are given as follows: dµ t = λ µ ( θ µ − µ t ) dt + (cid:0) α t ( h φ ) − µ t (cid:1) dW t + (cid:0) α t ( h φ ) − β t µ t (cid:1) dW t ,dβ t = (cid:18) λ β β t + ρ ( λ β + λ µ ) ρ µ t − ρλ µ θ µ ρ (cid:19) dt + (cid:0) α t ( h φ ) − µ t β t (cid:1) dW t + (cid:16) α t ( h φ ) − β t (cid:17) dW t . Numerically, in order to simulate α t , we can use the approximation scheme developed by Gobetet al [ ? ] or the extended Kalman filter studied by Pardoux [ ? , Chap.6].Also we consider another example for which we can apply proposition (4.3): the stochasticfactor Log Ornstein-Uhlenbeck model. the special features of this model is not only we can applyproposition (4.3), but also we are in a particular case of the signal-observation system (A.1)where A, B and G are deterministic. So we are in the framework of the classical Kalman-Bucyfilter with correlation between the signal and the observation processes, see Pardoux [ ? , Chap.6]and Kallianpur[ ? , Theo 10.5.1]. This filter is deduced from the general Kushner-Stratonovichequation (A.2), but the advantage of this filter is that it is a finite dimensional filter. Finite dimensional filter: stochastic factor Log Ornstein-Uhlenbeck modelLet us consider the following Log Ornstein-Uhlenbeck model: dS t S t = e V t (cid:0) µ t dt + dW t (cid:1) (4.11) dV t = λ V ( θ − V t ) dt + σ V ρdW t + σ V p − ρ dW t (4.12) dµ t = λ µ ( θ µ − µ t ) dt + σ µ dW t . (4.13)Then from the definition of ˜ µ t and ˜ β t and Itô’s formula, the risks of the system have the followingdynamics: d (cid:18) ˜ µ t ˜ β t (cid:19) = (cid:18) A ( t ) (cid:18) ˜ µ t ˜ β t (cid:19) + b ( t ) (cid:19) dt + G ( t ) d W t W t ! + B ( t ) W t W t ! . Here: A = − λ µ ρ [ λ µ − λ V ] ρ − λ V , b = λ µ θ µ − ρρ λ µ θ µ ! , G = σ µ − ρρ σ µ ! B = − ρρ λ V − λ V ! . where ρ = p − ρ .Therefore using theorem . . in [ ? ], we can deduce the following stochastic differentialequations for the filters: 12 (cid:18) µ t β t (cid:19) = (cid:18) A ( t ) (cid:18) µ t β t (cid:19) + b ( t ) (cid:19) dt + ( B ( t ) + Θ t ) d W t W t ! . (4.14)Where Θ t is the conditional covariance matrix ( × ) of the signal satisfies the following deter-ministic matrix Ricatti equation: d Θ t = A Θ t + Θ t A T + GG T − Θ t Θ Tt − Θ t B T − B Θ t . (4.15)Also we can consider the case where the mean θ of the stochastic volatility V t is a linearfunction of µ t . For example, assume the above dynamics of ( S t , V t , µ t ) with θ = µ t . Therefore,the filters estimate ( µ t , β t ) verifies (4.14). Here G and B are the same matrix given above, but A and b are given by: A = − λ µ ρ [ λ µ − λ V ] ρ − λ µ λ V σ V ρ − λ V , b = λ µ θ µ λ V σ V ρ − ρρ λ µ θ µ . Remark 4.8.
Also, we have the above results about the filters estimate if we consider the Stien-stein model, where the stock has the dynamics: dS t S t = | V t | (cid:0) µ t dt + dW t (cid:1) and the stochasticvolatility V t and the drift µ t are given by (4.12) and (4.13). Before presenting our results, let us recall that the trader’s objective is to solve the followingoptimization problem: J ( x ) = sup π ∈A t E [ U ( R πT )] x > , (5.1)where the dynamics of R πt in the full information context is given by: dR πt = R πt π t (cid:16) g ( V t ) µ t dt + g ( V t ) dW t (cid:17) . Here A t is the set of admissible controls π t which are F S -adapted process,take their value in acompact U ⊂ R , and satisfies the integrability condition: Z Tt g ( V s ) π s < ∞ P a.s. (5.2)We have showed that using the nonlinear filtering theory, the partial observation portfolioproblem is transformed into a full observation one with the additional filter in the dynamic ofthe wealth, for which one may apply the martingale or PDE approach.Here we will interested by the martingale approach in order to resolve our optimization problem.The motivation to use the martingale approach instead of the PDE approach is that we don’tneed to impose any constraint on the admissible control (see remark 5.16).As the reduced market model is not complete, due to the stochastic factor V , we have to solve therelated dual optimization problem. For that, we complement the martingale approach by usingthe PDE approach in order to solve explicitly the dual problem. For the case of CARA’s utilityfunctions, show by verification result, that under some assumptions on the market coefficients,the dual value function and the dual optimizer are related to the solution of a semilinear partialdifferential equation. 13 .1 Martingale approach Before presenting our result concerning the solution of the dual problem, let us begin by remind-ing some general results about the martingale approach.The martingale approach in incomplete market is based on a dual formulation of the optimizationproblem in terms of a suitable family of ( P , G ) -local martingales. The important result for thedual formulation is the martingale representation theorem given in [ ? ] for ( P , G ) -local martingaleswith respect to the innovation processes W and W . Lemma 5.1 (Martingale representation theorem) . Let A be any ( P , G ) -local martingale. Then,there exist a G -adapted processes φ and ψ , P a.s. square-integrable and such that A t = Z t φ s dW s + Z t ψ s dW s . (5.3)Now, we aim to describe the dual formulation of the optimization problem. We now make thefollowing assumption which will be useful in the sequel: Z T µ t dt < ∞ , Z T ν t dt < ∞ P − a.s. (5.4)For any G -adapted process ν = { ν t , ≤ t ≤ T } , which satisfies (5.4), we introduce the ( P , G ) -local martingale strictly positive: Z νt = exp (cid:18) − Z t µ s dW s − Z t ν s dW s − Z t µ s ds − Z t ν s ds (cid:19) (5.5)When, E [ Z νT ] = 1 , the process Z is a martingale and then there exists a probability measure Q equivalent to P with: d Q d P | G t = Z νT . Here µ is the risk related to the asset’s Brownian motion W , which is chosen such that Q isa equivalent martingale measure, that is, the process Z ν R is a ( P , G ) -local martingale. On theother hand, ν is the risk related to the stochastic volatility’s Brownian motion and this risk willbe determined as the optimal solution of the dual problem defined below.Consequently, from Itô’s formula, the process Z ν satisfies: dZ νt = − Z νt (cid:16) µ s dW s + ν s dW s (cid:17) . (5.6)As shown by Karatzas et al [ ? ], the solution of the primal problem (5.1) relying upon solvingthe dual optimization problem: J dual ( z ) = inf Q ∈Q E (cid:20) ˜ U ( z d Q d P ) (cid:21) := inf ν ∈K E h ˜ U ( zZ νT ) i , z > (5.7)Where: • Q is the set of equivalent martingale measures given by: Q = { Q ∼ P | R is a local ( Q , G ) − martingale } . (5.8)14 ˜ U is the convex dual of U given by: ˜ U ( y ) = sup m> [ U ( m ) − ym ] , m > . (5.9) • K is the Hilbert space of G -adapted process ν such that E (cid:20)Z T | ν t | dt (cid:21) < ∞ .We henceforth impose the following assumptions on the utility functions in order to guaranteethat the dual problem admits a solution ˜ ν ∈ K : Assumption 2. • For some p ∈ (0 , , γ ∈ (1 , ∞ ) , we have pU ′ ( x ) ≥ U ′ ( γx ) ∀ x ∈ (0 , ∞ ) . • x → xU ′ ( x ) is nondecreasing on (0 , ∞ ) . • For every z ∈ (0 , ∞ ) , there exists ν ∈ K such that ˜ J ( z ) < ∞ . By same arguments as in theorem . in Karatzas et al [ ? ], we have existence to the dualproblem (5.7). Proposition 5.2.
Under assumption 2, for all z > , the dual problem (5.7) admits a solution ˜ ν ( z ) ∈ K In the sequel, we denote by I :]0 , ∞ [ → ]0 , ∞ [ the inverse function of U ′ on ]0 , ∞ [ . It’s a decreasingfunction and verifies lim x → I ( x ) = ∞ and lim x →∞ I ( x ) = 0 .Now from Karatzas et al [ ? ] and Owen [ ? ], we have the following result about the solution ofthe primal utility maximization problem (2.4). Theorem 5.3.
The optimal wealth for the utility maximization problem (2.4) is given by ˜ R t = E (cid:20) Z ˜ νT Z ˜ νt I ( z x Z ˜ νT ) |G t (cid:21) where ˜ ν = ˜ ν ( z x ) is the solution of the dual problem and z x is the Lagrange multiplier such that E (cid:2) Z ˜ νT I ( z x Z ˜ νT ) (cid:3) = x . Also the optimal portfolio ˜ π is implicitly determined by the equation d ˜ R t = ˜ π t g ( V t ) d ˜ W t . (5.10) Remark 5.4.
The constraint E (cid:2) Z ˜ νT I ( z x Z ˜ νT ) (cid:3) = x to choose z x is satisfied if z x ∈ argmin z> { J dual ( z ) + xz } . (5.11)Now we begin by presenting our results about the solution of the dual problem. We remark from theorem 5.3 that optimal wealth depends on the optimal choice of ν . So weare interested in the following by finding the optimal risk ν which is solution of (5.7).Here we present two cases. Firstly, we show that in the case when the filter estimate of the pricerisk µ t ∈ F ˜ W t , the infimum of the dual problem is reached for ˜ ν = 0 . Secondly, for the generalcase, the idea is to derive a Hamilton-Jacobi-Bellman equation for dual problem, which involvesthe volatility risk ν as control process. 15 emma 5.5. Assume that µ t ∈ F ˜ W t , then the infimum of the dual problem is reached for ˜ ν = 0 ,that is: J dual ( z ) = inf Q ∈Q E (cid:20) ˜ U ( z d Q d P ) (cid:21) = E h ˜ U (cid:0) zZ T (cid:1)i . (5.12) Proof.
See Appendix A.Generally, the filter estimate of the price risk doesn’t satisfy lemma 5.5 and therefore it’s adifficult problem to derive an explicit characterization for the solution of the dual problem andtherefore for the optimal wealth and portfolio. For that, we need to present the dual problemas a stochastic control problem with controlled process Z νt and control process ν .Firstly, from the underlying dynamics of Z νt , we notice that our optimization problem . hasthree state variables which will be take in account to describe the associated Hamilton-Jaccobi-Belleman equation: the dynamic (5.6) of Z νt , the dynamic of the stochastic volatility ( V t ) whichis given in system ( Q ) and the dynamic of the filter estimate of the price risk µ t . Remark 5.6.
We have showed in filtering section, that the filter estimate µ t satisfies a stochas-tic differential equation which in general is infinite dimensional and is not a Markov process.Therefore, we can’t use it to describe the HJB. On the other hand, we have also showed thatfor some models of stochastic volatility models, we can obtain a finite dimensional stochasticdifferential equation for µ t which is also a Markov process. So in the sequel, we will assume thatthe filter µ t is Markov.On the other hand, we need in general to take in account the dynamics of µ t and β t . But forsimplification, we will consider β t as a linear function of µ t or a constant. Also for this choiceof β t , we can obtain, due to the separation technique used in proposition (5.17), a closed formfor the value function and the optimal portfolio. In the following, we assume that µ t is Markov. So for initial time t ∈ [0 , T ] and for fixed z ,the dual value function is defined by the following stochastic control problem: ˜ J ( z, t, z, v, m ) := inf ν ∈K E h ˜ U ( zZ νT ) | Z νt = z, V t = v, µ t = m i . (5.13)Where the dynamics of ( Z νt , V t , µ t ) are given as follows: dZ νt = − Z νt µ s dW s − Z νt ν s dW s dV t = f ( µ t , V t ) dt + ρk ( V t ) dW t + p − ρ k ( V t ) dW t dµ t = τ ( µ t ) dt + ϑ ( µ t ) dW t + Υ( µ t ) dW t . where f is a linear function.Remark that the dual value function in (5.7) is simply deduced from J dual ( z ) = J dual ( z, , z, v, m ) .If we assume that Y t = ( V t , µ t ) be a bi-dimensional process, then the controlled process ( Z νt , Y t ) satisfies the following dynamics: dZ νt = − Z νt ψ ( Y s ) dW s − Z νt ν s dW s (5.14) dY t = Γ( Y t ) dt + Σ( Y t ) dW t (5.15)where W t = ( W t , W t ) is a bi-dimensional Brownian motion, and for y = ( v, m ) , we have: ψ ( y ) = m, Γ( y ) = (cid:18) f ( m, v ) τ ( m ) (cid:19) Σ( y ) = (cid:18) ρk ( v ) p − ρ k ( v ) ϑ ( m ) Υ( m ) (cid:19) . Then we have the new reformulation of the above stochastic problem ( ?? ) and its HJBequation as follows: J dual ( z, t, z, y ) := inf ν ∈K E h ˜ U ( zZ νT ) | Z νt = z, Y t = y i . (5.16)Now assuming that ˜ U satisfies the following property: ˜ U ( λx ) = g ( λ ) ˜ U ( x ) + g ( λ ) , (5.17)for λ > and for any functions g and g .The special advantage of this assumption is: we can solve the dual problem (5.7) independentlyof z . In general, a solution to the dual problem (5.7) depends on z , but for this type of ˜ U thisdependence vanishes. Then (5.7) reads as follows: J dual ( z ) = g ( z ) inf ν ∈K E h ˜ U ( Z νT ) | Z νt = z, Y t = y i + g ( z ) . Let us now denote ˜ J ( t, z, y ) = inf ν ∈K E h ˜ U ( Z νT ) | Z νt = z, Y t = y i . (5.18)Remark that the solution of the dual problem (5.7) is given by: J dual ( z ) := J dual ( z, , z, v, m ) = g ( z ) ˜ J (0 , z, y ) + g ( z ) . (5.19)Formally, the Hamilton-Jacobi-Bellman equation associated to the above stochastic control prob-lem (5.18) is the following nonlinear partial differential equation: ∂ ˜ J∂ t + 12 T r (cid:16) Σ( y )Σ T ( y ) D y ˜ J (cid:17) + Γ T ( y ) D y ˜ J + inf ν ∈K (cid:20)
12 ( ψ ( y ) + ν ) z D z ˜ J − z [ ψ ( y ) K T ( y ) + νK T ] D z,y ˜ J (cid:21) = 0 . (5.20)with the boundary condition ˜ J ( T, x, y ) = U ( x ) . (5.21)And the associated optimal dual optimizer ˜ ν is given by: ˜ ν t = K T D z,y ˜ Jz D z ˜ J .
Here D y and D y denote the gradient and the Hessian operators with respect to the variable y . D z,y is the second derivative vector with respect to the variables z and y and for y = ( v, m ) , K ( y ) = (cid:18) ρk ( v ) ϑ ( m ) (cid:19) and K ( y ) = (cid:18) p − ρ k ( v )Υ( m ) (cid:19) . The above HJB is nonlinear, but if we consider the case of CARA’s utility functions and via asuitable transformation, we can make this equation semilinear and then characterize the dualvalue function ˜ J through the classical solution of this semilinear equation which is more simplerthan the usual fully nonlinear HJB equation. 17 .2 Special cases for utility function Let us consider the two more standard utility functions: logarithmic and power, defined by: U ( x ) = ln( x ) x ∈ R + x p p x ∈ R + , p ∈ (0 , For these functions, the convex dual functions associated are given by: ˜ U ( z ) = − (1 + ln( z )) z ∈ R + − z q q z ∈ R , q = pp − These utility functions are of particular interests: firstly, they satisfy property (5.17)and sec-ondly, due to the homogeneity of the convex dual functions together with the fact that theprocess Z νt and the control ν appear linearly, we can suggest a suitable transformation, forwhich we can characterize the dual value functions ˜ J through a classical solution of a semilinearsemilinear partial differential equations which will be described below.Let us now make some assumptions which will be useful for proving our verification results. Assumption (H) i ) Γ and Σ are Lipscitz and C with bounded derivatives. ii ) ΣΣ T is uniformly elliptic, that is, there exists c > such that for y, ξ ∈ R : X i,j =1 (ΣΣ T ( y )) ij ξ i ξ j ≥ c | ξ | .iii ) Σ is bounded or is a deterministic matrix. iv ) There exists a positive constant ǫ such that exp (cid:18) ǫ Z T ( ψ ( Y t ) + ν t ) dt (cid:19) ∈ L ( P ) . Notice that the Lipschitz assumption on Γ and Σ ensure the existence and uniqueness of thesolution of (5.15). Moreover, we have: E [ sup ≤ s ≤ t | Y s | ] < ∞ . (5.22) For the logarithmic utility case, we can look for a candidate solution of ( ?? ) and ( ?? ) in theform : ˜ J ( t, z, y ) = − (1 + ln ( z )) − Φ( t, y ) (5.23)Then direct substitution of (5.23) in (5.20) and (5.21) gives us the following semilinear partialdifferential equation for Φ : − ∂ Φ ∂ t − T r (cid:0) Σ( y )Σ T ( y ) D y Φ (cid:1) + H ( y, D y Φ) = 0 , (5.24)18ith the boundary condition: Φ( T, y ) = 0 . (5.25)Where the Hamiltonian H is defined by: H ( y, Q ) = − Γ T ( y ) Q + inf ν (cid:18)
12 ( ψ ( y ) + ν ) (cid:19) . We now state a verification result for the logarithmic case, which relates the solution of theabove semilinear (5.24) and (5.25) to the stochastic control problem (5.18).
Theorem 5.7 (verification theorem) . Let assumption H i ) holds. Suppose that there existsa solution Φ ∈ C , ([0 , T ) × R ) ∩ C ([0 , T ] × R ) to the semilinear (5.24) with the terminalcondition (5.25). Also we assume that Φ satisfies a polynomial growth conditon, i.e: | Φ( t, y ) | ≤ C (1 + | y | k ) for some k ∈ N . Then, for all ( t, z, y ) ∈ [ O, T ] × R + × R ˜ J ( t, z, y ) ≤ − − ln( z ) − Φ( t, y ) , and for the optimal risk ˜ ν = 0 , we have ˜ J ( t, x, y ) = − − ln ( x ) − Φ( t, y ) .Proof. Let ˜ J ν ( t, z, y ) = E h ˜ U ( Z νT ) | Z νt = z, Y t = y i . From (5.18) and ˜ U ( z ) = − − ln( z ) , we havethe following expression for ˜ J ν : ˜ J ν ( t, z, y ) = − − ln( z ) + E (cid:20) Z Tt ( ψ ( Y s ) + ν s ) ds (cid:21) . (5.26)Let ν be an arbitrary control process, Y the associated process with Y t = y and define thestopping time θ n := T ∧ inf { s > t : | Y s − y | ≥ n } . Now, let Φ be a C , solution to (5.24). Then, by Itôs formula, we have: Φ( θ n , Y θ n ) = Φ( t, y ) + Z θ n t (cid:18) ∂ Φ ∂ t + 12 T r (ΣΣ T D y Φ) + Γ T D y Φ (cid:19) ( s, Y s ) ds + Z θ n t (( D y Φ) T Σ)( s, Y s ) dW s ≤ Φ( t, y ) + 12 Z θ n t ( ψ ( Y s ) + ν s ) ds + Z θ n t (( D y Φ) T Σ)( s, Y s ) dW s (5.27)From the definition of θ n , the integrand in the stochastic integral is bounded on [ t, θ n ] , a con-sequence of the continuity of D y Φ and assumption H i ) . Then, by taking expectation, oneobtains: E [Φ( θ n , Y θ n )] ≤ Φ( t, y ) + E (cid:20) Z θ n t ( ψ ( Y s ) + ν s ) ds (cid:21) . We now take the limit as n increases to infinity, then θ n → T a.s . From the growth conditionsatisfied by Φ and ( ?? ), we can deduce the uniform integrability of (Φ( θ n , Y θ n )) n . Therefore, itfollows from the dominated convergence theorem and the boundary condition (5.25) that for all ν ∈ K : − Φ( t, y ) ≤ E (cid:20) Z Tt ( ψ ( Y s ) + ν s ) ds (cid:21) Then from (5.23), we have: ˜ J ν ( t, z, y ) ≤ − − ln( z ) − Φ( t, y ) . ν by ˜ ν = 0 which is the optimal risk, we canfinally deduce that: ˜ J ˜ ν ( t, z, y ) = − − ln( z ) − Φ( t, y ) . which ends the proof since ˜ J ( t, z, y ) = inf ν ∈K ˜ J ν ( t, z, y ) Let us now study the regularity of the solution Φ to the semilinear (5.24) with the terminalcondition (5.25). Proposition 5.8.
Under assumptions H i ) and ii ) , there exists a solution Φ ∈ C , ([0 , T ) × R ) ∩ C ([0 , T ] × R ) with polynomial qrowth condition in y , to the semilinear (5.24) with theterminal condition (5.25).Proof. Under assumptions i ) and ii ) and the fact that the Hamiltonian H satisfies a globalLipschitz condition on D y Φ , we can deduce from theorem . in Fleming and soner [ ? , p.163]the existence and uniqueness of a classical solution to the semilinear equation(5.24). As the above reasons given in the logarithmic case, we can suggest that the value function mustbe of the form: ˜ J ( t, z, y ) = − z q q exp( − Φ( t, y )) . (5.28)Then if we substitute the above form in (5.20)and (5.21), we can deduce the following semilinearP.D.E for Φ : − ∂ Φ ∂ t − T r (cid:0) ΣΣ T D y Φ (cid:1) + H ( y, D y Φ) = 0 , (5.29) Φ( T, y ) = 0 . (5.30)The Hamiltonian H is defined by: H ( y, Q ) = 12 Q T Σ( y )Σ T ( y ) Q − Q T Γ( y ) + inf ν ∈K (cid:20) q ( q − ψ ( y ) + ν ) + q (cid:0) ψ ( y ) K T + νK T (cid:1) Q (cid:21) (5.31) = 12 Q T (cid:0) Σ( y )Σ T ( y ) − G ( y ) (cid:1) Q − Q T F ( y ) + Ψ( y ) . (5.32)where for y := ( v, m ) : G ( y ) = qq − K ( y ) K T ( y ) F ( y ) = Γ( y ) − qψ ( y ) K Ψ( y ) = 12 q ( q − ψ ( y ) . We now state a verification result for the power case, which relates the solution of the abovesemilinear (5.29) and (5.30) to the stochastic control problem (5.18).20 heorem 5.9 (verification theorem) . Let assumptions H i ) , iii ) and iv ) hold. Suppose thatthere exists a solution Φ ∈ C , ([0 , T ) × R ) ∩ C ([0 , T ] × R ) with linear growth condition onthe derivation D y Φ , to the semilinear (5.29) with the terminal condition (5.30). Then, for all ( t, x, y ) ∈ [ O, T ] × R + × R i) ˜ J ( t, z, y ) ≤ − z q q exp( − Φ( t, y )) .Now, assume that there exists a minimizer ˜ ν of ν −→ q ( q − ν + qνK ( y ) T D y Φ such that − ∂ Φ ∂ t − T r (cid:0) ΣΣ ∗ D y Φ (cid:1) + H ( y, D y Φ) = 0 . Thenii) ˜ J ( t, z, y ) = − z q q exp( − Φ( t, y )) .and the associated optimal ˜ ν is given by the Markov control { ˜ ν t = ˜ ν ( t, Y t ) } with ˜ ν t = − q − K T ( Y T ) D y Φ( t, Y t ) . (5.33) Proof.
Let us introduce the new probability Q ν as follows: d Q ν d P = exp (cid:18) − Z t qψ ( Y u ) dW u − Z t qν u dW u − Z t q ( ψ ( Y u ) + ν u ) du (cid:19) , From assumption iv ) the probability measure Q ν with the density process d Q ν d P is well defined,see Liptser and Shiryaev [ ? , P.233].Let ˜ J ν ( t, z, y ) = E h ˜ U ( Z νT ) | Z νt = z, Y t = y i .From (5.18) and ˜ U ( z ) = − z q q , we have from Itô’s formula the following expression for ˜ J ν : ˜ J ν ( t, z, y ) = − z q q E ν (cid:20) exp (cid:18)Z Tt q ( q − ψ ( Y u ) + ν ) du (cid:19) | Y t = y (cid:21) . (5.34)Also by Girsanov’s theorem, the dynamics of Y under Q ν , is given by: dY t = (Γ( Y t ) − qψ ( Y t ) K ( Y t ) − qν t K ( Y t )) dt + Σ( Y t ) dW νt , (5.35)where W ν is a bi-dimensional Brownian motion under Q ν .Now, let Φ be a C , solution to (5.29), then by Itô’s formula applied to Φ( t, Y t ) under Q ν , oneobtains: Φ( θ n , Y T ) = Φ( t, y ) + Z Tt (cid:18) ∂ Φ ∂ t + (Γ − qψK − qν t K ) T D y Φ + 12
T r (ΣΣ T D y Φ) (cid:19) ( u, Y u ) du + Z Tt ( D Ty Φ Σ)( u, Y u ) dW νu Φ is solution of (5.29), then one obtains: Φ( T, Y T ) = Φ( t, y ) + Z Tt (cid:16) H ( y, D y Φ) + (Γ − qψK − qν t K ) T D y Φ (cid:17) ( u, Y u ) du + Z Tt ( D Ty Φ Σ)( u, Y u ) dW νu (5.36) ≤ Φ( t, y ) + Z Tt q ( q − ψ ( Y u ) + ν ) du + 12 Z Tt (cid:0) D Ty Φ ΣΣ T D y Φ (cid:1) ( u, Y u ) du + Z Tt ( D Ty Φ Σ)( u, Y u ) dW νu , (5.37)where the inequality comes from the representation (5.31) of the Hamiltonian.Therefore, we have: exp( − Φ( t, y )) E ν (cid:20) exp (cid:18) − Z Tt (cid:0) D Ty Φ ΣΣ T D y Φ (cid:1) ( u, Y u ) du − Z Tt ( D Ty Φ Σ)( u, Y u ) dW νu (cid:19)(cid:21) ≤ E ν (cid:20)Z Tt q ( q − ψ ( Y u ) + ν ) du (cid:21) . Let us now consider the exponential Q ν -local martingales: ǫ πt = exp (cid:18) − Z t ( D Ty Φ Σ)( u, Y u ) dW νu − Z t (cid:0) D Ty Φ ΣΣ T D y Φ (cid:1) ( u, Y u ) du (cid:19) . From the Lipschitz condition assumed in i ) and from iii ) , we can deduce from Gronwall’s lemmathat there exists a positive constant C such that: | Y t | ≤ C (cid:18) Z t | W νu | du + | W νt | (cid:19) Then we deduce that there exists some ǫ > such that sup t ∈ [0 ,T ] E ν [exp( ǫ | Y t | )] < ∞ . (5.38)Therefore from (5.38) and the fact that D y Φ satisfies a linear growth condition in y, we candeduce that ǫ π is a martingale under Q ν , therefore we have: exp( − Φ( t, y )) ≤ E ν (cid:20)Z Tt q ( q − ψ ( Y u ) + ν ) du (cid:21) . The above inequality is proved for all ν ∈ K , therefore we can deduce from (5.34) that: ˜ J ( t, z, y ) ≤ − z q q exp( − Φ( t, y )) . since ˜ J ( t, z, y ) = inf ν ∈K ˜ J ν ( t, z, y ) , then i ) is proved.Now by repeating the above argument and observing that the control ˜ ν given by (5.33),achieves equality in (5.37), we can finally deduce that: ˜ J ν ( t, z, y ) = − z q q exp( − Φ( t, y )) . Also since ˜ J ( t, z, y ) = inf ν ∈K ˜ J ν ( t, z, y ) , then ii ) is proved.22e now study the existence of a classical solution to (5.29)-(5.30).In fact, the existence of a classical solution to (5.29)-(5.30) cannot be found directly in theliterature since Q → H ( y, Q ) is not globally Lipschitz on Q but satisfies a quadratic growthcondition on Q . For that we can use the approach taken in [ ? ] by considering a certain sequenceof approximating P.D.Es which are the HJB-equations of certain stochastic control problems forwhich the existence of smooth solution is well-known.Let us make some assumptions which will be useful to prove the regularity for the solutionof (5.29). Assumption (H’)
Let us consider either one of the following conditions: I ) -If Σ is a deterministic matrix: In this case we need the following assumption: i ) Γ and ψ are Lipschitz and C with bounded derivatives. II ) -If Σ is not a deterministic matrix: In this case we need the following assumptions: i ) Γ and ψ.K are Lipschitz and C . ii ) ψ , K K T are C with bounded derivatives. iii ) ΣΣ T − qq − K K T is uniformly elliptic.By the similar arguments used by Pham in [ ? ] and from the standard verification theorem provedby Fleming and soner [Theorem 3.1 P.163][ ? ], we can deduce our regularity result for the casewhen the Hamiltonian is not globally Lipschitz but satisfies a quadratic growth condition. Theorem 5.10.
Under one the assumptions ( H’ ), there exists a solution Φ ∈ C , ([0 , T ) × R ) ∩ C ([0 , T ] × R ) with linear growth condition on the derivation D y Φ , to the semilinear (5.29) withthe terminal condition (5.30). Remark 5.11.
In general, a closed form solution to (5.29) with the terminal condition (5.30)does not exist. But we show that for some stochastic volatility model and in the case when thefilters estimate are Gaussian, we can obtain a closed form, see section .
Let us now describe the relation between the optimal trading strategy and the optimal dualoptimiser.
We have showed from theorem 5.3, that the optimal wealth, and by consequence the optimalportfolio, depend on the optimal dual optimiser ˜ ν . So we will study this relation in the specialcase of utility functions studied above.From theorem 5.3, we have: ˜ R t = E (cid:20) Z ˜ νT Z ˜ νt I ( z x Z ˜ νT ) |G t (cid:21) (5.39)where ˜ ν is the optimal dual maximizer and z x is the Lagrange multiplier such that E (cid:2) Z ˜ νT I ( z x Z ˜ νT ) (cid:3) = x . Before presenting our result concerning the optimal wealth and the optimal portfolio, inorder to avoid any confusion, let us describe the dynamics of the wealth R t in terms of theprocess Y t := ( V t , µ t ) as follows: dR t = R t π t ( ψ ( Y t ) δ ( Y t ) dt + δ ( Y t ) dW t ) (5.40)23here ψ ( Y t ) = µ t and δ ( Y t ) = g ( V t ) . Logarithmic utility: U ( x ) = ln( x ) . Proposition 5.12.
We suppose that the assumptions of theorems 5.7 and 5.8 hold. Then theoptimal wealth process is given by ˜ R t = xZ t . Also the optimal portfolio ˜ π and the primal valuefunction are given by: ˜ π t = ψ ( Y t ) δ ( Y t ) := µ t g ( V t ) and J ( x ) = ln( x ) − Φ(0 , Y ) . (5.41) where Φ is the solution of the semilinear equation (5.24) with boundary condition (5.25).Proof. In this case we have I ( x ) = 1 x and from theorem ?? , the dual optimizer ˜ ν = 0 . Moreover,the Lagrange multiplier z x = 1 x . Therefore from (5.39), the optimal wealth is given by ˜ R t = xZ t . (5.42)By applying Itô’s formula to (5.42) and from proposition 3.2, we obtain that: d ˜ R t = ˜ R t ψ ( Y t ) d ˜ W t On the other hand, we have from (5.40) that d ˜ R t = ˜ R t ˜ π t δ ( Y t ) d ˜ W t . Therefore comparing thesetwo expressions for ˜ R t , we obtain that the optimal portfolio ˜ π is given by (5.41).Finally fromthe definition of the primal value function and (5.42), we have J ( x ) = ln( x ) − E [ln( Z T )] =ln( x ) + 1 + ˜ J (0 , , Y ) = ln( x ) − Φ(0 , Y ) . The last equality comes from theorem 5.7. Power utility: U ( x ) = x p /p < p < . Proposition 5.13.
We suppose the assumptions of theorems 5.9 and 5.10 hold. Then the optimalwealth is given by: ˜ R t = x E [( Z ˜ νT ) q ] ( Z ˜ νt ) q − exp ( − Φ( t, Y t )) . the associated optimal portfolio is given by the Markov control { ˜ π t = ˜ π ( t, Y t ) } with ˜ π t = 11 − p ψ ( Y t ) δ ( Y t ) − K T ( Y t ) δ ( Y t ) D y Φ( t, Y t ) (5.43) and the primal value function is given by: J ( x ) = x p p exp( − (1 − p )Φ(0 , Y )) . Where q = pp − , ˜ ν is given by (5.33) and Φ is a solution of the semilinear equation (5.29) withboundary condition (5.21).Proof. In this case we have I ( x ) = x / ( p − and from theorem ?? , the dual optimizer ˜ ν is givenby (5.33). The Lagrange multiplier z x = x E [ Z p/p − T ] ! p − . Therefore from (5.39), the optimal24ealth is given by ˜ R t = E (cid:20) Z ˜ νT Z ˜ νt I ( z x Z ˜ νT ) |G t (cid:21) = E (cid:20) Z ˜ νT Z ˜ νt ( z x ) / ( p − ( Z ˜ νT ) / ( p − |G t (cid:21) = x E [( Z ˜ νT ) q ] 1 Z ˜ νt E (cid:2) ( Z ˜ νT ) q |G t (cid:3) Therefore from theorem 5.9, we deduce that: ˜ R t = x E [( Z ˜ νT ) q ] ( Z ˜ νt ) q − exp ( − Φ( t, Y t )) . (5.44)Now, as in the logarithmic case, by writing d ˜ R t = ˜ R t π t δ ( V t ) d ˜ W t and applying Itô’s formula to ( Z ˜ νt ) q − exp ( − Φ( t, Y t )) , then after comparing the two expressions for ˜ R t , we deduce that: ˜ π t = 11 − p ψ ( Y t ) δ ( Y t ) − K T ( Y t ) δ ( Y t ) D y Φ( t, Y t ) . Finally, from (5.44) and the boundary condition Φ( T, Y t ) = 0 , we have: J ( x ) = x p p E [( Z ˜ νT ) q ] − p = x p p exp( − (1 − p )Φ(0 , Y )) . where the last equality comes from theorem 5.9.Let us now deduce the following relation between the primal and dual control function. Corollary 5.14.
The optimal portfolio ˜ π is given by ˜ π t = 11 − p ψ ( Y t ) δ ( Y t ) − − p K T ( K T ) − δ ( Y t ) ˜ ν t . (5.45) Proof.
The proof can be deduced easily from theorem ?? and proposition 5.13. Remark 5.15.
For the logarithmic case, we notice that in the case of partial information,the optimal portfolio can be formally derived from the full information case by replacing theunobservable risk premium ˜ µ t by its estimate µ t . But on the other hand, in the power utilityfunction, this property does not hold and the optimal strategy cannot be derived from the fullinformation case by replacing the risk ˜ µ t by its best estimate µ t due to the last additional termwhich depend on the filter.This property corresponds to the so called separation principle. It is proved in Kuwana [ ? ] thatcertainty equivalence holds if and only if the utilities functions are logarithmic. Remark 5.16.
The advantage of using the martingale approach instead of the dynamic program-ming approach (PDE approach) is that we don’t need to impose any constraint on the admissibleportfolio controls, while it is essential in the case of the PDE approach. In fact, with the PDEapproach, we need to make the following constraint on the admissible portfolio controls: sup t ∈ [0 ,T ] E [exp( c | δ ( Y t ) π t | )] < ∞ , for some c > . (5.46) this constraint is indispensable to impose in order to show a verification theorem in the case ofpower utility function. .4 Application Here we give an example of stochastic volatility model for which we can obtain a closed formfor the value function and the optimal portfolio. Let us consider the Log Ornstein-Uhlenbeckmodel defined in (4.11), (4.12) and (4.13). Also we consider the power utility function U ( x ) = x p p , < p < .Firstly, notice that we have the following dynamics of ( R πt , V t , µ t ) in the full observation frame-work: dR πt = R πt π t (cid:16) µ t e V t dt + e V t dW t (cid:17) dV t = λ V ( θ − V t ) dt + σ V ρdW t + σ V p − ρ dW t dµ t = ( − λ µ µ t + λ µ θ µ ) dt + Θ t dW t + Θ t dW t . where the last dynamics is deduced from (4.14). Θ and Θ are solutions of Riccati equation(4.15).Therefore the primal value function J ( x ) and the associated optimal portfolio ˜ π t are givenexplicitly. Proposition 5.17.
The optimal portfolio is given by: ˜ π t = 1 p − µ t e V t − ρσ V e V t [ ˜ A ( t ) + ( T − t )] + Θ e V t (2 A ( t ) µ t + B ( t )) . and the primal value function is given by: J ( x ) = x p p exp (cid:2) − (1 − p ) (cid:0) ˜ A (0) V + ˜ B (0) − V T − A (0) µ − B (0) µ − C (0) (cid:1)(cid:3) . where: ˜ A ( t ) = − λ V Z Tt ( T − s ) e − λ V ( s − t ) ds. ˜ B ( t ) = Z Tt h −
12 ( σ V − qq − − ρ ) σ V ) A ( s ) + (cid:18) ( σ V − qq − − ρ ) σ V ) + λ V θ (cid:19) A ( s ) −
12 ( σ V − qq − − ρ ) σ V )( T − s ) − λ V θ ( T − s ) i ds. and A is solution of the following Riccati equation: A ′ ( t ) = − (cid:18) Θ + Θ − qq − (cid:19) A ( t ) + 2( λ µ + q Θ ) A ( t ) − q ( q − , with A ( T ) = 0 and B ( t ) = Z Tt B ( s ) A ( s ) exp (cid:2) − ( λ µ + q Θ )( s − t ) + 2(Θ + Θ − qq − ) Z st A ( u ) du i ds.C ( t ) = Z Tt h (Θ + Θ ) A ( s ) + 12 (cid:18) Θ + Θ − qq − (cid:19) B ( s ) − B ( s ) B ( s ) i ds. where B ( s ) = +2 (cid:20) ( ρσ V Θ + p − ρ σ V Θ − qq − p − ρ σ V Θ )( ˜ A ( s ) − ( T − s )) − λ µ θ µ (cid:21) and with terminal conditions: ˜ A ( T ) = ˜ B ( T ) = A ( T ) = B ( T ) = C ( T ) = 0 .Proof. See Appendix A. 26
Appendix
Filtering
Let us consider the following partially observation system: dX t = A ( X t ) dt + G ( X t ) dM t + B ( X t ) dW t (A.1) dY t = dW t + h ( X t ) dt (A.2)Here X is the two dimensional signal process and Y is the two dimensional observation process. A is a × matrix, G, B are × matrix and h is × matrix . W and M are two dimensionalindependents Brownian motions.Now, we will be interested in the filtering problem which consists in evaluating the conditionalexpectation of the unobservable process having the observations. In the sequel, we denote thisconditional expectation by α t ( φ ) = E (cid:2) φ ( X t ) |F Yt (cid:3) , where F Y is the filtration generated by theobservation process Y .Then one of the approaches to obtain the evolution equation for α t is to change the measure.Using the change of measure ˜ P given in (3.4), we can define a new measure ˜ P , such that theobservation process becomes a ˜ P Brownian motion independent of the signal variable X t . Forthat we need to discuss some conditions under which the process L is a martingale: L t = exp − X i =1 Z t h i ( X s ) dW is − X i =1 Z t h i ( X s ) ds ! . (A.3)Firstly, the classical condition is Novikov’s condition: E (cid:20) exp (cid:18) Z t h ( X s ) ds + 12 Z t h ( X s ) ds (cid:19)(cid:21) < ∞ . Normally Novikov’s condition is quite difficult to verify directly, so we need to use an alternativeconditions under which the process L is a martingale.From lemma . in [ ? ], we can deduce that L is a martingale if the following conditions aresatisfied: E (cid:20)Z t ( || h ( X s ) || ) ds (cid:21) < ∞ , E (cid:20)Z t L s || h ( X s ) || ds (cid:21) < ∞ ∀ t > . (A.4)Let us now denote by Λ t the (cid:16) ˜ P , F (cid:17) -martingale given by Λ t = L t . We then have: d P d ˜ P |F t = Λ t , ≤ t ≤ T = exp X i =1 Z t h i ( X s ) dW is − X i =1 Z t h i ( X s ) ds ! . Therefore the computation of α t ( φ ) is obtained by the so-called Kallianpur-Striebel formula,which is related to Bayes formula. For every φ ∈ B ( R d ) , we have the following representation: α t ( φ ) := E (cid:2) φ ( X t ) |F Yt (cid:3) = ˜ E (cid:2) φ ( X t )Λ t |G Yt (cid:3) ˜ E (cid:2) Λ t |G Yt (cid:3) := ψ t ( φ ) ψ t (1) , (A.5)with ψ t ( φ ) := ˜ E [ φ ( X t )Λ t |G Yt ] is the unnormalized conditional distribution of φ ( X t ) , given G Yt , ψ t (1) can be viewed as the normalising factor and B ( R d ) is the space of bounded measurablefunctions R → R . 27n the following, we assume that for all t ≥ , ˜ P (cid:20)Z t [ ψ s ( || h || )] ds < ∞ (cid:21) = 1 , for all t > . (A.6)Let us now introduce the following notations which will be useful in the sequel. Notations 2.
Let K = 12 ( BB T + GG T ) and A be the generator associated with the process X in the second order differential operator: A φ = X i,j =1 K ij ∂ x i x j φ + X i =1 A i ∂ x i φ, for φ ∈ B ( R d ) . (A.7) and its adjoint A ∗ is given by: A ∗ φ = X i =1 ∂ x i x j ( K ij φ ) − X i =1 ∂ x i ( A i φ ) . (A.8) Also we introduce the following operator B = ( B k ) k =1 : B k φ = X i =1 B ik ∂ x i φ, for φ ∈ B ( R d ) . (A.9) and the adjoint of the operator B is given by B k, ∗ = ( B k, ∗ ) k =1 : B , ∗ φ = − X i =1 ∂ x i ( B i φ ) , B , ∗ φ = − X i =1 ∂ x i ( B i φ ) . (A.10)The following two propositions show that the unnormalized conditional distribution (resp. theconditional distribution) of the signal is a solution of a linear stochastic partial differentialequation often called the Zakai equation (resp. nonlinear stochastic and parabolic type partialdifferential equation often called the Kushner-Stratonovich equation). These results due to Bainand Crisan [ ? ] and Pardoux [ ? ]. Proposition A.1.
Assume that the signal and observation processes satisfy (A.1) and (A.2).If conditions (A.4) and (A.6) are satisfied then the unnormalized conditional distribution ψ t satisfies the following Zakai equation: dψ t ( φ ) = ψ t ( Aφ ) dt + ψ t (cid:0)(cid:0) h + B (cid:1) φ (cid:1) d ˜ W t + ψ t (cid:0)(cid:0) h + B (cid:1) φ (cid:1) d ˜ W t . (A.11) for any φ ∈ B ( R ) . Proposition A.2.
Assume that the signal and observation processes satisfy (A.1) and (A.2). Ifconditions (A.4) and (A.6) are satisfied then the conditional distribution α t satisfies the followingKushner-Stratonovich equation: dα t ( φ ) = α t ( Aφ ) dt + (cid:2) α t (cid:0)(cid:0) h + B (cid:1) φ (cid:1) − α t ( h ) α t ( φ ) (cid:3) dW t + (cid:2) α t (cid:0)(cid:0) h + B (cid:1) φ (cid:1) − α t ( h ) α t ( φ ) (cid:3) dW t . (A.12) for any φ ∈ B ( R ) . roof of lemma 5.5 From equation (5.5), the definition of the conditional expectation and Jensen’s inequality, itfollows for any ν ∈ K : E h ˜ U ( zZ νT ) i = E (cid:20) E (cid:20) ˜ U (cid:18) z exp (cid:18) − Z T µ s dW s − Z T µ s ds − Z T ν s dW s − Z T ν s ds (cid:19)(cid:19) |F ˜ W T (cid:21)(cid:21) ≥ E (cid:20) ˜ U (cid:18) z exp (cid:18) − Z T µ s dW s − Z T µ s ds (cid:19) E (cid:20) exp (cid:18) − Z T ν s dW s − Z T ν s ds (cid:19) |F ˜ W T (cid:21)(cid:19)(cid:21) . On the other hand, E (cid:20) exp (cid:18) − Z T ν s dW s − Z T ν s ds (cid:19)(cid:21) = 1 a.s. In fact, from the definitionof the conditional expectation, it remains to prove that for each positive function h , for each, t , ......t k ∈ [0 , T ] , we have: E (cid:20) exp (cid:18) − Z T ν s dW s − Z T ν s ds (cid:19) h (cid:16) ˜ W t , ..... ˜ W t k (cid:17)(cid:21) = E h h (cid:16) ˜ W t , ..... ˜ W t k (cid:17)i . As ν is a G -adapted, we can define a new probability measure P ν equivalent to P on G T givenby: d P ν d P = exp (cid:18) − Z T ν u dW − Z T ν u du (cid:19) By Girsanov theorem, N is a G Brownian motion under P ν . On the other hand, from thedynamic of ˜ W given by d ˜ W = dN t + µ t dt and the assumption that µ t ∈ F ˜ W t , we deduce thatthe law of ˜ W remains the same under P and P ν . Thus: E ν h h (cid:16) ˜ W t , ..... ˜ W t k (cid:17)i := E (cid:20) exp (cid:18) − Z T ν s dW s − Z T ν s ds (cid:19) h (cid:16) ˜ W t , ..... ˜ W t k (cid:17)(cid:21) = E h h (cid:16) ˜ W t , ..... ˜ W t k (cid:17)i . Therefore E (cid:20) exp (cid:18) − Z T ν s dW s − Z T ν s ds (cid:19)(cid:21) = 1 and then one obtains: E h ˜ U ( zZ νT ) i ≥ E (cid:20) ˜ U (cid:18) z exp (cid:18) − Z t µ s dW s − Z t µ s ds (cid:19)(cid:19)(cid:21) := E h ˜ U (cid:0) zZ T (cid:1)i . On the other hand, we have from the definition of the dual problem that ˜ J ( z ) ≤ E h ˜ U (cid:0) zZ T (cid:1)i ,so we conclude that J dual ( z ) = E h ˜ U (cid:0) zZ T (cid:1)i . Proof of proposition 5.17
With the Log-Ornstein model given by (4.11),(4.12) and (4.13), the assumptions H and H’ i ) hold. Therefore from proposition 5.13, we have ˜ π t = 1 p − µ t e V t − K T ( Y t ) e V t D y Φ( t, Y t ) , K T = ( ρσ V Θ ) and Φ is solution of (5.29). Generally, equation (5.29) does not haveclosed-form, but with this model we can deduce a closed form for Φ by using the followingseparation transformation: For y = ( v, m ) , Φ( t, y ) = ˜Φ( t, v ) − ˜ f ( t, v, m ) . The general idea of this separation transformation has been used by a lot of authors likeFleming [ ? ] Pham [ ? ], Rishel[ ? ].., in order to express the value function in terms of the solutionto a semilinear parabolic equation.Now, substituting the above form of Φ into (5.29) gives us: − ∂ ˜Φ ∂ t + ∂ ˜ f∂ t − σ V " ∂ ˜Φ ∂ v − ∂ ˜ f∂ v + (cid:16) ρσ V Θ + p − ρ σ V Θ (cid:17) ∂ ˜ f∂ v,m + 12 (Θ + Θ ) ∂ ˜ f∂ m + 12 ( σ V − qq − − ρ ) σ V ) " ( ∂ ˜Φ ∂ v ) − ∂ ˜Φ ∂ v ∂ ˜ f∂ v + ( ∂ ˜ f∂ v ) − ( λ V ( θ − v ) − qmρσ V ) ∂ ˜Φ ∂ v − ∂ ˜ f∂ v ! + ( − λ µ m + λ µ θ µ − qm Θ ) ∂ ˜ f∂ m + 12 (cid:18) Θ + Θ − qq − (cid:19) ( ∂ ˜ f∂ m ) + 12 q ( q − m − (cid:18) ρσ V Θ + p − ρ σ V Θ − qq − p − ρ σ V Θ (cid:19) ( ∂ ˜ f∂ m ∂ ˜Φ ∂ v − ∂ ˜ f∂ m ∂ ˜ f∂ v ) = 0 . Thus we have a coupled PDEs for which we have not able to find its solution in general. Thekey is to separate the considered PDE into a PDE in ˜Φ and another in ˜ f , with the fact that ˜Φ( T, v ) = 0 and ˜ f ( T, v, m ) = 0 . These two last conditions come from the boundary condition(5.21).But, there is also another difficult to obtain a explicit solution for ˜ f . This difficulty comesfrom the terms ∂ ˜ f∂ v,m and ∂ ˜ f∂v ∂ ˜ f∂m . For that we need to impose the following separation form on ˜ f : ˜ f ( t, v, m ) = v. ( T − t ) + f ( t, m ) , with f ( T, m ) = 0 .Finally, we have the following PDEs for ˜Φ and f for which we can deduce an explicit form asfollows: − ∂ ˜Φ ∂ t − σ V ∂ ˜Φ ∂ v + 12 ( σ V − qq − − ρ ) σ V )( ∂ ˜Φ ∂ v ) − (cid:18) ( σ V − qq − − ρ ) σ V ) + λ V ( θ − v ) (cid:19) ∂ ˜Φ ∂ v − λ V ( T − t ) v + 12 ( σ V − qq − − ρ ) σ V )( T − t ) + λ V θ ( T − t ) . (A.13)and ∂f∂ t + 12 (Θ + Θ ) ∂ f∂ m + 12 (cid:18) Θ + Θ − qq − (cid:19) ( ∂ ˜ f∂ m ) + " − ( ρσ V Θ + p − ρ σ V Θ − qq − p − ρ σ V Θ )( ∂ ˜Φ ∂ v − ( T − t )) − λ µ m + λ µ θ µ − qm Θ ∂f∂ m + 12 q ( q − m + qmρσ V ∂ ˜Φ ∂v − qmρσ V ( T − t ) (A.14)Notice that the PDE for f depends on ∂ ˜Φ ∂ v , but we show below that the solution of the PDFsatisfied by ˜Φ is polynomial of degree , then by deriving it, we obtain a term which does not30epend on v . So we have a PDE for f which depends only on m , therefore an explicit form canbe deduced.The solution of (A.13) with the boundary condition ˜Φ( T, v ) = 0 is given by: ˜Φ( t, v ) = ˜ A ( t ) v + ˜ B ( t ) where: ˜ A t and ˜ B ( y ) are respectively solutions of the following differential equations: ˜ A ′ ( t ) = λ V ˜ A ( t ) − λ V ( T − t )) , with ˜ A ( T ) = 0 , ˜ B ′ ( t ) = 12 ( σ V − qq − − ρ ) σ V ) A ( t ) − (cid:18) ( σ V − qq − − ρ ) σ V ) + λ V θ (cid:19) A ( t )+ 12 ( σ V − qq − − ρ ) σ V )( T − t ) + λ V θ ( T − t ) with ˜ B ( T ) = 0 , One easily verifies that ˜ A ( t ) , ˜ B ( t ) given in proposition 5.17 are solutions of the above differ-ential equations.On the other hand, the solution of (A.14) with the boundary condition f ( T, m ) = 0 is given by: f ( t, m ) = A ( t ) m + B ( t ) m + C ( t ) Where: A ′ ( t ) = − (cid:18) Θ + Θ − qq − (cid:19) A ( t ) + 2( λ µ + q Θ ) A ( t ) − q ( q − ,B ′ ( t ) = (cid:20) − + Θ − qq − ) A ( t ) + ( λ µ + q Θ ) (cid:21) B ( t ) − qρσ V ˜ A ( t ) + qρσ V ( T − t )+ 2 (cid:20) ( ρσ V Θ + p − ρ σ V Θ − qq − p − ρ σ V Θ )( ˜ A ( t ) − ( T − t )) − λ µ θ µ (cid:21) A ( t ) ,C ′ ( t ) = − (Θ + Θ ) A ( t ) − (cid:18) Θ + Θ − qq − (cid:19) B ( t )+ (cid:20) ( ρσ V Θ + p − ρ σ V Θ − qq − p − ρ σ V Θ )( ˜ A ( t ) − ( T − t )) − λ µ θ µ (cid:21) B ( t ) . with terminal condition A ( T ) = B ( T ) = C ( T ) = 0 . The solution of the riccati equation satisfiedby A ( t ) can be deduced from [ ? ]. For B ( t ) and C ( t ) , on easily verifies that their expressionsgiven in proposition 5.17 are solutions of the above differential equations.Finally, from proposition 5.13 and the above solutions of ˜Φ and ff