Stochastic Verification Theorem of Forward-Backward Controlled Systems for Viscosity Solutions
aa r X i v : . [ m a t h . O C ] M a r Stochastic Verification Theorem of Forward-BackwardControlled Systems for Viscosity Solutions
Liangquan ZHANG , ∗
1. School of mathematics, Shandong University, China.2. Laboratoire de Mathématiques,Université de Bretagne Occidentale,29285 Brest Cédex, France.
Abstract
In this paper, we investigate the controlled systems described by forward-backwardstochastic differential equations with the control contained in drift, diffusion and gen-erator of BSDEs. A new verification theorem is derived within the framework of vis-cosity solutions without involving any derivatives of the value functions. It is worthto pointing out that this theorem has wider applicability than the restrictive classicalverification theorems. As a relevant problem, the optimal stochastic feedback controlsfor forward-backward systems are discussed as well.
Key words:
Stochastic optimal control, forward-backward stochastic differential equa-tions, H-J-B equations, viscosity solutions, super/sub-differentials, optimal feedback con-trols.
Since the fundamental work of Pardoux & Peng [1], the theory of BSDEs and FBSDEshave become a powerful tool in many fields, such as mathematics finance, optimal control,stochastic games, partial differential equations and homogenization etc. Recently, the par-tially coupled FBSDEs controlled systems have been studied in [2], [3], and [4], where theauthors used the dynamic programming principle and proved that the value function is tobe the unique viscosity solution of the H-J-B equations. In [5], the authors investigated the ∗ Corresponding author. E-mail: [email protected]. This work was supported by Marie CurieInitial Training Network (ITN) project: ”Deterministic and Stochastic Controlled System and Applica-tion”, FP7-PEOPLE-2007-1-1-ITN, No. 213841-2 and National Natural Science Foundation of China Grant10771122, Natural Science Foundation of Shandong Province of China Grant Y2006A08 and National BasicResearch Program of China (973 Program, No. 2007CB814900). R n the space of n -dimensional Euclidean space,by R n × d the space the matrices with order n × d , by S n the space of symmetric matriceswith order n × n . h· , ·i and |·| denote the scalar product and norm in the Euclidean space,respectively. * appearing in the superscripts denoted the transpose of a matrix.Let T > and let (Ω , F , P ) be a complete probability space, equipped with a d -dimensional standard Brownian motion { W ( t ) } ≤ t ≤ T . For a given s ∈ [ t, T ] , we supposethat the filtration {F st } s ≤ t ≤ T is generated as the following F st = σ { W ( r ) − W ( s ) ; s ≤ r ≤ T } ∨ N , where N contains all P -null sets in F . In particular, if s = 0 we write F t = F st . Let X be a Hilbert space with the norm k·k X , and p, ≤ p ≤ + ∞ , define the set L p F ( a, b ; X ) = { φ ( · ) = { φ ( t, ω ) : a ≤ t ≤ b }| φ ( · ) is an F t -adapted, X -valued measurableprocess on [ a, b ] , and E R ba k φ ( t, ω ) k p X d t < + ∞ . } . Let U is a given closed set in some Euclidean space R m . For a given s ∈ [0 , T ] , we denoteby U ad ( s, T ) the set of U -valued F st -predictable processes. For any initial time s ∈ [ t, T ] and initial state y ∈ R d , we consider the following stochastic control systems d X s,y ; u ( t ) = b ( t, X s,y ; u ( t ) , u ( t )) d t + σ ( t, X s,y ; u ( t ) , u ( t )) d W t , d Y s,y ; u ( t ) = − f ( t, X s,y ; u ( t ) , Y s,y ; u ( t ) , Z s,y ; u ( t ) , u ( t )) d t + Z s,y ; u ( t ) d W t ,X s,y ; u ( s ) = x, Y s,y ; u ( T ) = Φ ( X s,y ; u ( T )) . (1.1)where b : R d × U → R d ,σ : R d × U → R d × d ,f : [0 , T ] × R d × R × R d × U → R , Φ : R d → R . They satisfy the following conditions(H1) b and σ are continuous in t. (H2) For some L > , and all x, x ′ ∈ R d , v, v ′ ∈ U, a.s. (cid:12)(cid:12)(cid:12) b ( t, x, v ) − b (cid:16) t, x ′ , v ′ (cid:17)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) σ ( t, x, v ) − σ (cid:16) t, x ′ , v ′ (cid:17)(cid:12)(cid:12)(cid:12) ≤ L (cid:16)(cid:12)(cid:12)(cid:12) x − x ′ (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) v − v ′ (cid:12)(cid:12)(cid:12)(cid:17) . v ( · ) ∈ U ad , the first control system of(1.1) has a unique strong solution { X s,y ; u ( t ) , ≤ s ≤ t ≤ T } . (H3) f and Φ are continuous in t. (H4) For some L > , and all x, x ′ ∈ R d , y, y ′ ∈ R ,z, z ′ ∈ R d , v, v ′ ∈ U, a.s. (cid:12)(cid:12)(cid:12) f ( t, x, y, z, v ) − f (cid:16) t, x ′ , y ′ , z ′ , v ′ (cid:17)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) Φ ( x ) − Φ (cid:16) x ′ (cid:17)(cid:12)(cid:12)(cid:12) ≤ L (cid:16)(cid:12)(cid:12)(cid:12) x − x ′ (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) y − y ′ (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) z − z ′ (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) v − v ′ (cid:12)(cid:12)(cid:12)(cid:17) . From the classical theory of BSDEs, we claim that there exists a triple ( X s,y ; u , Y s,y ; u , Z ts,y ; u ) , which is the unique solution of the FBSDEs (1.1).Given a control process u ( · ) ∈ U ad ( s, T ) we consider the following cost functional J ( s, y ; u ( · )) = Y s,y ; u ( s ) , ( s, y ) ∈ [0 , T ] × R d , (1.2)where the process Y s,y ; u is defined by FBSDEs (1.1). It follows from the uniqueness of thesolution of the SDEs and BSDEs that Y s,y ; u ( s + δ )= Y s + δ,X s,y ; u ( t + δ ); u ( s + δ )= J ( t + δ, X s,y ; u ( t + δ )) , a.s.The object of the optimal control problem is to minimize the cost function J ( s, y ; u ( · )) ,for a given ( s, y ) ∈ [0 , T ] × R d , over all u ( · ) ∈ U ad ( s, T ) . We denote the above problem by C s,y to recall the dependence on the initial time s and the initial state y . The value functionis defined as V ( s, y ) = inf u ( · ) ∈U ad ( s,T ) J ( s, y ; u ( · )) . (1.3)An admissible pair ( X ⋆ ( · ) , u ⋆ ( · )) is called optimal for C s,y if u ⋆ ( · ) achieves the minimumof J ( s, y ; u ( · )) over U ad ( s, T ) . As we have known that the verification technique plays an important role in testingfor optimality of a given admissible pair and, especially, in constructing optimal feedbackcontrols. Let us recall the similar classical verification theorem as follows.
Theorem 1.
Let W ∈ C , (cid:0) [0 , T ] × R d (cid:1) be a solution of the following Hamiliton-Jacobi-Bellman (H-J-B) equations: (cid:26) ∂∂t W ( t, x ) + H ( t, x, W, DW, D W ) = 0 , ( t, x ) ∈ [0 , T ] × R d ,W ( T, x ) = Φ ( x ) , x ∈ R d . (1.4)3 he Hamilitonian is given by H (cid:0) t, x, W, DW, D W (cid:1) = inf u ∈ U H (cid:0) t, x, W, DW, D W, u (cid:1) , where H (cid:0) t, x, Ψ , D Ψ , D Ψ , u (cid:1) = 12 tr (cid:0) σσ ∗ ( t, x, u ) D Ψ (cid:1) + h D Ψ , b ( t, x, u ) i + f ( t, x, Ψ ( t, x ) , D Ψ ( t, x ) · σ ( t, x, u ) , u ) , ( t, x, u ) ∈ [0 , T ] × R d × U, Ψ ∈ C , (cid:0) [0 , T ] × R d (cid:1) . Here the function b, σ, f and Φ are supposed to satisfy (H1)-(H4). Then1 ◦ ) W ( s, y ) ≤ J ( s, y ; u ( · )) for any ( s, y ) ∈ [0 , T ] × R d and u ( · ) ∈ U ad ( s, T ) . ◦ ) Supposed that a given admissible pair ( x ⋆ ( · ) , u ⋆ ( · )) , here x ⋆ ( · ) = X ⋆ ( · ) , for theproblem C s,y satisfies ∂∂t W ( t, x ⋆ ( t ))+ H (cid:0) t, x ⋆ ( t ) , W ( t, x ⋆ ( t )) , DW ( t, x ⋆ ( t )) , D W ( t, x ⋆ ( t )) , u ⋆ ( t ) (cid:1) = 0 , P -a.s., a.e. t ∈ [ s, T ] ; (1.5) then ( x ⋆ ( · ) , u ⋆ ( · )) is an optimal pair for the problem C s,y . The proof follows from Theorem 9 in Section 3 in our paper.
Remark 2.
By H-J-B equations, (1.5) is equivalent to the following form min u ∈ U H (cid:0) t, x ⋆ ( t ) , W ( t, x ⋆ ( t )) , DW ( t, x ⋆ ( t )) , D W ( t, x ⋆ ( t )) , u (cid:1) = H (cid:0) t, x ⋆ ( t ) , W ( t, x ⋆ ( t )) , DW ( t, x ⋆ ( t )) , D W ( t, x ⋆ ( t )) , u ⋆ ( t ) (cid:1) . Then, an optimal feedback control u ⋆ ( t, x ) can be constructed by minimizing H (cid:0) t, x, W ( t, x ) , DW ( t, x ) , D W ( t, x ) , u (cid:1) over u ∈ U. emark 3. We claim that (1.5) is equivalent to W ( s, y ) = J ( s, y ; u ⋆ ( · )) . Actually, we have
Φ ( X ⋆ ( T )) − W ( s, y )= W ( T, X ⋆ ( T )) − W ( s, y )= Z Ts dd t W ( t, x ⋆ ( t )) d t = Z Ts [ ∂∂t W ( t, x ⋆ ( t ))+ H (cid:0) t, x ⋆ ( t ) , W ( t, x ⋆ ( t )) , DW ( t, x ⋆ ( t )) , D W ( t, x ⋆ ( t )) , u ⋆ ( t ) (cid:1) − f ( t, x ⋆ ( t ) , W ( t, x ) , DW ( t, x ⋆ ( t )) · σ ( t, x ⋆ ( t ) , u ⋆ ( t )) , u ∗ ( t ))] d t + Z Ts W x ( t, x ⋆ ( t )) · σ ( t, x ⋆ ( t ) , u ⋆ ( t )) d W t ] , which implies W ( s, y )= J ( s, y ; u ⋆ ( · )) + Z Ts [ ∂∂t W ( t, x ⋆ ( t ))+ H (cid:0) t, x ⋆ ( t ) , W ( t, x ⋆ ( t )) , DW ( t, x ⋆ ( t )) , D W ( t, x ⋆ ( t )) , u ⋆ ( t ) (cid:1) ] d t It is necessary to point out that in Theorem 1 we need W ∈ C , (cid:0) [0 , T ] × R d (cid:1) . However,when we take the verification function W to be the value function V, as V satisfies the HJBequations if V ∈ C , (cid:0) [0 , T ] × R d (cid:1) . Unfortunately, in general the H-J-B equations (1.4)do not admit smooth solutions, which makes the applicability of the classical verificationtheorem very restrictive and is a major deficiency in dynamic programming theory. As wehave known that the viscosity theory of nonlinear PDEs was launched by Crandall andLions. In this theory, all the derivatives involved are replaced by the super-differentials andsub-differentials, and solution in viscosity sense can be only continuous function (For moreinformation see in [10]). Besides, since the verification theorems can be played primary rolesin constructing optimal feedback controls, while in many practical problems H-J-B equationsdo not admit smooth solutions, hence, we want to answer the question aforementioned.Our paper is organized as follows: In Section 2, we introduce some preliminary re-sults about viscosity solutions and the associated the second order one-sided super/sub-differentials. In Section 3, a new verification theorem in term of viscosity solutions and thesuper-differentials are established. At last, we show the way to find the optimal feedbackcontrols in Section 4. 5 Super-differentials, Sub-differentials, and Viscosity So-lutions
Let Q be an open subset of R n , and v : Q → R be a continuous function. Definition 4.
The second order one-sided super-differentials (resp., sub-differentials) of v at ( t , x ) ∈ [0 , T ) × R n , denoted by D + t + ,x v ( t , x ) (resp. D − t + ,x v ( t , x ) ), is a set defined by D + t + ,x v ( t , x )= { ( p, q, Θ) ∈ R × R d × S d (cid:12)(cid:12) lim t → t + ,x → x v ( t, x ) − v ( t , x ) − p ( t − t ) − h q, x − x i − ( x − x ) ∗ Q ( x − x ) | t − t | + | x − x | ≤ } .by (resp., D − t + ,x v ( t , x )= { ( p, q, Θ) ∈ R × R d × S d (cid:12)(cid:12) lim t → t + ,x → x v ( t, x ) − v ( t , x ) − p ( t − t ) − h q, x − x i − ( x − x ) ∗ Q ( x − x ) | t − t | + | x − x | ≥ } ). Let us recall the definition of a viscosity solution for (1.4) from [3] or [4]
Definition 5.
An continuous function v on [0 , T ] × R n is called a viscosity subsolution (resp.,supersolution) of the H-J-B equations (1.4) if v ( T, x ) ≤ Φ ( x ) . and ∂ϕ∂t ( t , x ) + inf u ∈ U (cid:8) H (cid:0) t , x , ϕ ( t , x ) , Dϕ ( t , x ) , D ϕ ( t , x ) , u (cid:1)(cid:9) ≥ ( ≤ ) 0 (2.1) whenever v − ϕ attains a local maximum (resp., minimum) at ( t , x ) in a right neighborhoodof ( t , x ) for ϕ ∈ C , ([0 , T ] × R n ) . A function v is called a viscosity solution of (1.4) if itis both a viscosity subsolution and a supersolution of (1.4). The equivalence of Definition 4 and the Definition 5 in which derivatives of test func-tions are replaced by elements of the second order one-sided sub- and super-differentials areestablished with the help of a well-known result that we present below and whose proof canbe found in [13]. 6 emma 6.
Let ( t , x ) ∈ [0 , T ] × R n be giveni) ( p, q, Θ) ∈ D + t + ,x v ( t , x ) if and only if there exists ϕ ∈ C , ([0 , T ] × R n ) satisfies (cid:18) ∂ϕ∂t ( t , x ) , D x ϕ ( t , x ) , D ϕ ( t , x ) (cid:19) = ( p ( t , x ) , q ( t , x ) , Θ ( t , x )) and such that v − ϕ achieves its maximum at ( t , x ) ∈ [0 , T ] × R n from right side on t .ii) ( p, q, Θ) ∈ D − t + ,x v ( t , x ) if and only if there exists ϕ ∈ C , ([0 , T ] × R n ) satisfies (cid:18) ∂ϕ∂t ( t , x ) , D x ϕ ( t , x ) , D ϕ ( t , x ) (cid:19) = ( p ( t , x ) , q ( t , x ) , Θ ( t , x )) and such that v − ϕ achieves its minimum at ( t , x ) ∈ [0 , T ] × R n from right side on t .Moreover, if v has polynomial growth, i.e., if | v ( t, x ) | ≤ C (cid:16) | x | k (cid:17) for some k ≥ , ( t, x ) ∈ [0 , T ] × R n , (2.2) then ϕ can be chosen so that ϕ, ϕ t , Dϕ, D ϕ satisfy (2.2) (with possibly different constants C ). Under the assumptions [H1]-(H4), we have the following results.
Lemma 7.
There exists a constant
C > such that, for all ≤ t ≤ T, x, x ′ ∈ R d , ( | V ( t, x ) ≤ C (1 + | x | ) | , (cid:12)(cid:12) V ( t, x ) − V (cid:0) t ′ , x ′ (cid:1)(cid:12)(cid:12) ≤ C (cid:16)(cid:12)(cid:12) t − t ′ (cid:12)(cid:12) + (cid:12)(cid:12) x − x ′ (cid:12)(cid:12)(cid:17) . (2.3) Moreover, V is a unique solution in the class of continuous functions which grow at mostpolynomially at infinity. The proof can be seen in [2] or [4]. Then according to Definition 5 and Lemma 6, wehave the following result.
Lemma 8.
We claim that inf ( p,q, Θ ,u ) ∈ D + t + ,x v ( t,x ) × U [ p + H ( t, x, v, q, Θ , u )] ≥ , ∀ ( t, x ) ∈ [0 , T ) × R d . (2.4) In this section, we give the stochastic verification theorem for Forward-Backward ControlledSystems within the framework of viscosity solutions. Firstly, we need the following twolemmas. 7 emma 9.
Suppose that (H1)-(H4) hold. Let ( s, y ) ∈ [0 , T ) × R d be fixed and let ( X s,y ; u ( · ) , u ( · )) be an admissible pair. Define processes z ( r ) . = b ( r, X s,y ; u ( r ) , u ( r )) ,z ( r ) . = σ ( r, X s,y ; u ( r ) , u ( r )) σ ∗ ( r, X s,y ; u ( r ) , u ( r )) ,z ( r ) . = f ( r, X s,y ; u ( r ) , Y s,y ; u ( r ) , Z s,y ; u ( r ) , u ( r )) . Then lim h → h Z t + ht | z i ( r ) − z i ( t ) | d r = 0 , a.e. t ∈ [0 , T ] , i = 1 , , . (3.1)The proof can be found in [7] or [13]. Lemma 10.
Let g ∈ C ([0 , T ]) . Extend g to ( −∞ , + ∞ ) with g ( t ) = g ( T ) for t > T, and g ( t ) = g (0) , for t < . Suppose that there is a integrable function ρ ∈ L (0 , T ; R ) and some h > , such that g ( t + h ) − g ( t ) h ≤ ρ ( t ) , a.e. t ∈ [0 , T ] , h ≤ h . Then g ( β ) − g ( α ) ≤ Z βα lim sup h → g ( t + h ) − g ( t ) h d r, ∀ ≤ α ≤ β ≤ T. Proof.
Applying Fatou’s Lemma, we have Z βα ρ ( r ) d r ≥ Z βα lim sup h → g ( r + h ) − g ( r ) h d r ≥ lim sup h → Z βα g ( r + h ) − g ( r ) h d r = lim sup h → R β + hα + h g ( r ) d r − R βα g ( r ) d rh = lim sup h → R β + hβ g ( r ) d r − R α + hα g ( r ) d rh = g ( β ) − g ( α ) . The main result in this section is the following.
Theorem 11. ( Verification Theorem ) Assume that (H1)-(H4) hold. Let v ∈ C (cid:0) [0 , T ] × R d (cid:1) , e a viscosity solution of the H-J-B equations (1.4), satisfying the following conditions: i) v ( t + h, x ) − v ( t, x ) ≤ C (1 + | x | m ) h, m ≥ , for all x ∈ R d , < t < t + h < T. ii) v is semiconcave , uniformly in t, i.e . there exists C ≥ such that for every t ∈ [0 , T ] , v ( t, · ) − C |·| is concave on R d (3.2) Then we have v ( s, y ) ≤ J ( s, y ; u ( · )) , for any ( s, y ) ∈ (0 , T ] × R d and any u ( · ) ∈ U ad ( s, T ) . (3.3) Forthurmore, let ( s, y ) ∈ (0 , T ] × R d be fixed and let (cid:0) X s,y ; u ( · ) , u ( · ) (cid:1) be an admissible pairfor Problem C sy such that there exist a function ϕ ∈ C , (cid:0) [0 , T ] ; R d (cid:1) and a triple (cid:0) p, q, Θ (cid:1) ∈ (cid:0) L F t ( s, T ; R ) × L F t (cid:0) s, T ; R d (cid:1) × L F t (cid:0) s, T ; S d (cid:1)(cid:1) (3.4) satisfying (cid:0) p ( t ) , q ( t ) , Θ ( t ) (cid:1) ∈ D + t + ,x v (cid:0) t, X s,y ; u ( t ) (cid:1) , (cid:0) ∂ϕ∂t (cid:0) t, X s,y ; u ( t ) (cid:1) , D x ϕ (cid:0) t, X s,y ; u ( t ) (cid:1) , D ϕ (cid:0) t, X s,y ; u ( t ) (cid:1)(cid:1) = (cid:0) p ( t ) , q ( t ) , Θ ( t ) (cid:1) ,ϕ ( t, x ) ≥ v ( t, x ) ∀ ( t , x ) = ( t, x ) , a.e. t ∈ [0 , T ] , P -a.s. (3.5) and E (cid:20)Z Ts (cid:2) p ( t ) + H (cid:0) t, X s,y ; u ( t ) , ϕ ( t ) , p ( t ) , Θ ( t ) , u ( t ) (cid:1)(cid:3) d t (cid:21) ≤ , (3.6) where ϕ ( t ) = ϕ (cid:0) t, X s,y ; u ( t ) (cid:1) . Then (cid:0) X s,y ; u ( · ) , u ( · ) (cid:1) is an optimal pair for the problem C sy . Proof.
Firstly, (3.3) follows from the uniqueness of viscosity solutions of the H-J-B equations(1.4). It remains to show that (cid:0) X s,y ; u ( · ) , u ( · ) (cid:1) is an optimal.We now fix t ∈ [ s, T ] such that (3.4) and (3.5) hold at t and (3.1) holds at t for z ( · ) = b ( · ) ,z ( · ) = σ ( · ) σ ( · ) ∗ z ( · ) = f ( · ) . We claim that the set of such points is of full measure in [ s, T ] by Lemma 9. Now we fix ω ∈ Ω such that the regular conditional probability P (cid:0) ·| F st (cid:1) ( ω ) , given F st is well defined.In this new probability space, the random variables X s,y ; u ( t ) , p ( t ) , q ( t ) , Θ ( t ) are almost surely deterministic constants and equal to X s,y ; u ( t , ω ) , p ( t , ω ) , q ( t , ω ) , Θ ( t , ω ) , W is still thea standard Brownian motion although now W ( t ) = W ( t , ω ) almost surely. The space isnow equipped with a new filtration {F sr } s ≤ r ≤ T and the control process u ( · ) is adapted tothis new filtration. For P -a.s. ω the process X s,y ; u ( · ) is a solution of (1.1) on [ t , T ] in (cid:0) Ω , F , P (cid:0) ·| F st (cid:1) ( ω ) (cid:1) with the inial condition X s,y ; u ( t ) = X s,y ; u ( t , ω ) . Then on the probability space (cid:0) Ω , F , P (cid:0) ·| F st (cid:1) ( ω ) (cid:1) , we are going to apply Itô’s formulato ϕ on [ t , t + h ] for any h > ,ϕ (cid:0) t + h, X s,y ; u ( t + h ) (cid:1) − ϕ (cid:0) t , X s,y ; u ( t ) (cid:1) = Z t + ht (cid:20) ∂ϕ∂t (cid:0) r, X s,y ; u ( r ) (cid:1) + (cid:10) D x ϕ (cid:0) r, X s,y ; u ( r ) (cid:1) , b ( r ) (cid:11) + 12 tr (cid:8) σ ( r ) ∗ D xx ϕ (cid:0) r, X s,y ; u ( r ) (cid:1) σ ( r ) (cid:9)(cid:21) d r + Z t + ht (cid:10) D x ϕ (cid:0) r, X s,y ; u ( r ) (cid:1) , σ ( r ) (cid:11) d W r . Taking conditional expectation value E F st ( · ) ( ω ) , dividing both sides by h , and using (3.5),we have h E F st ( ω ) (cid:2) v (cid:0) t + h, X s,y ; u ( t + h ) (cid:1) − v (cid:0) t , X s,y ; u ( t ) (cid:1)(cid:3) ≤ h E F st ( ω ) (cid:2) ϕ (cid:0) t + h, X s,y ; u ( t + h ) (cid:1) − ϕ (cid:0) t , X s,y ; u ( t ) (cid:1)(cid:3) = 1 h E F st ω (cid:26)Z t + ht (cid:20) ∂ϕ∂t (cid:0) r, X s,y ; u ( r ) (cid:1) + (cid:10) D x ϕ (cid:0) r, X s,y ; u ( r ) (cid:1) , b ( r ) (cid:11) + 12 tr (cid:8) σ ( r ) ∗ D xx ϕ (cid:0) r, X s,y ; u ( r ) (cid:1) σ ( r ) (cid:9)(cid:21) d r (cid:27) (3.7)Letting h → , and employing the similar delicate method as in the proof of Theorem4.1 of Gozzi et al. [12], we have h lim sup h → E F st ( ω ) (cid:2) v (cid:0) t + h, X s,y ; u ( t + h ) (cid:1) − v (cid:0) t , X s,y ; u ( t ) (cid:1)(cid:3) ≤ ∂ϕ∂t (cid:0) t , X s,y ; u ( t , ω ) (cid:1) + (cid:10) D x ϕ (cid:0) t , X s,y ; u ( t , ω ) (cid:1) , b ( t ) (cid:11) + 12 tr (cid:8) σ ( t ) ∗ D xx ϕ (cid:0) t , X s,y ; u ( t , ω ) (cid:1) σ ( t ) (cid:9) = p ( t , ω ) + (cid:10) q ( t , ω ) , b ( t ) (cid:11) + 12 tr (cid:8) σ ( t ) ∗ Θ ( t , ω ) σ ( t ) (cid:9) By (3.2), we know, from [12], that there exist ρ ∈ L ( t , T ; R ) and ρ ∈ L (Ω; R ) E (cid:20) h (cid:2) v (cid:0) t + h, X s,y ; u ( t + h ) (cid:1) − v (cid:0) t, X s,y ; u ( t ) (cid:1)(cid:3)(cid:21) ≤ ρ ( t ) , for h ≤ h , for some h > , (3.8)and E F st ( ω ) (cid:20) h (cid:2) v (cid:0) t + h, X s,y ; u ( t + h ) (cid:1) − v (cid:0) t, X s,y ; u ( t ) (cid:1)(cid:3)(cid:21) ≤ ρ ( ω ) , for h ≤ h , for some h > . (3.9)holds, respectively. By virtue of Fatou’s Lemma, noting (3.9), we obtain lim sup h → h E (cid:2) v (cid:0) t + h, X s,y ; u ( t + h ) (cid:1) − v (cid:0) t , X s,y ; u ( t ) (cid:1)(cid:3) = lim sup h → h E h E F st ( ω ) (cid:8) v (cid:0) t + h, X s,y ; u ( t + h ) (cid:1) − v (cid:0) t , X s,y ; u ( t ) (cid:1)(cid:9)i ≤ E (cid:20) lim sup h → h E F st ( ω ) (cid:8) v (cid:0) t + h, X s,y ; u ( t + h ) (cid:1) − v (cid:0) t , X s,y ; u ( t ) (cid:1)(cid:9)(cid:21) ≤ E (cid:20) p ( t ) + (cid:10) q ( t ) , b ( t ) (cid:11) + 12 tr (cid:8) σ ( t ) ∗ Θ ( t ) σ ( t ) (cid:9)(cid:21) , (3.10)for a.e. t ∈ [ s, T ] . Then the rest of the proof goes exactly as in [11]. We apply Lemma 10to g ( t ) = E (cid:2) v (cid:0) t, X s,y ; u ( t ) (cid:1)(cid:3) , using (3.8), (3.6) and (3.10) to get E (cid:2) v (cid:0) T, X s,y ; u ( T ) (cid:1) − v ( s, y ) (cid:3) ≤ E (cid:26)Z Ts (cid:20) p ( t ) + (cid:10) q ( t ) , b ( t ) (cid:11) + 12 tr (cid:2) σ ( t ) ∗ Θ ( t ) σ ( t ) (cid:3) d t (cid:21)(cid:27) ≤ − E (cid:20)Z Ts f ( t ) d t (cid:21) . From this we claim that v ( s, y ) ≥ E (cid:20) v (cid:0) T, X s,y ; u ( T ) (cid:1) + Z Ts f ( t ) d t (cid:21) = E (cid:20) Φ (cid:0) X s,y ; u ( T ) (cid:1) + Z Ts f ( t ) d t (cid:21) . Thus, combining the above with the first assertion (3.3), we prove the (cid:0) X s,y ; u ( · ) , u ( · ) (cid:1) is anoptimal pair. The proof is complete. 11 emark 12. The condition (3.6) is just equivalent to the following: p ( t ) = min u ∈ U H (cid:0) t, X s,y ; u ( t ) , ϕ ( t ) , q ( t ) , Θ ( t ) , u (cid:1) = H (cid:0) t, X s,y ; u ( t ) , ϕ ( t ) , q ( t ) , Θ ( t ) , u ( t ) (cid:1) , a.e. t ∈ [ s, T ] , P -a.s., (3.11) where ϕ ( t ) is defined in Theorem 11. This is easily seen by recalling the fact that v is theviscosity solution of (1.4): p ( t ) + min u ∈ U H (cid:0) t, X s,y ; u ( t ) , ϕ ( t ) , q ( t ) , Θ ( t ) , u (cid:1) ≥ , which yields (3.11) under (3.6). In this section, we describe the method to construct optimal feedback controls by the ver-ification Theorem 11 obtained. First, let us recall the definition of admissible feedbackcontrols.
Definition 13.
A measurable function u from [0 , T ] × R d to U is called an admissible feedbackcontrol if for any ( s, y ) ∈ [0 , T ) × R d there is a weak solution X s,y ; u ( · ) of the following SDEs: d X s,y ; u ( t ) = b ( t, X s,y ; u ( t ) , u ( t )) d t + σ ( t, X s,y ; u ( t ) , u ( t )) d W ( t ) , d Y s,y ; u ( t ) = − f ( t, X s,y ; u ( t ) , Y s,y ; u ( t ) , u ( t )) d t + d M s,y ; u ( t ) ,X s,y ; u ( s ) = x, Y s,y ; u ( T ) = Φ ( X s,y ; u ( T )) , (4.1) where M s,y ; u is an R -valued F s,y ; u -adapted right continuous and left limit martingale van-ishing in t = 0 which is orthogonal to the driving Brownian motion W. Here F s,y ; u = (cid:0) F X s,y ; u t (cid:1) t ∈ [ s,T ] is the smallest filtration and generated by X s,y ; u , which is such that X s,y ; u is F s,y ; u -adapted. Obviously, M s,y ; u is a part of the solution of BSDEs of (4.1). Simultane-ously, we suppose that f satisfies the Lipschitz condition. (cid:12)(cid:12)(cid:12) f ( t, x, y, u ) − f (cid:16) t, x ′ , y ′ , u ′ (cid:17)(cid:12)(cid:12)(cid:12) ≤ L (cid:16)(cid:12)(cid:12)(cid:12) x − x ′ (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) y − y ′ (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) u − u ′ (cid:12)(cid:12)(cid:12)(cid:17) x, x ′ ∈ R d , y, y ′ ∈ R , u, u ′ ∈ U. An admissible feedback control u ⋆ is called optimal if ( X ⋆ ( · ; s, y ) , u ⋆ ( · , X ⋆ ( · ; s, y ))) isoptimal for the problem C s,y for each ( s, y ) is a solution of (4.1) corresponding to u ⋆ . Theorem 14.
Let u ⋆ be an admissible feedback control and p ⋆ , q ⋆ , and Θ ⋆ be measurablefunctions satisfying ( p ⋆ ( t, x ) , q ⋆ ( t, x ) , Θ ( t, x )) ∈ D + t + ,x V ( t, x ) (4.2)12 or all ( t, x ) ∈ [0 , T ] × R d . If p ⋆ ( t, x ) + H ( t, x, V ( t, x ) , q ⋆ ( t, x ) , Θ ⋆ ( t, x ) , u ⋆ ( t, x ))= inf ( p,q, Θ ,u ) ∈ D + t + ,x V ( t,x ) × U [ p + H ( t, x, V ( t, x ) , q, Θ , u )]= 0 (4.3) for all ( t, x ) ∈ [0 , T ] × R d , then u ⋆ is optimal. P roof
From Theorem 11, we get the desired result. ✷ Remark 15.
Actually, it is fairly easy to check that in Eq.(4.1), Y s,y ; u ( · ) is determinedby ( X s,y ; u ( · ) , u ( · )) . Hence, we need to investigate the conditions imposed in Theorem 11 toensure the existence and uniqueness of X s,y ; u ( · ) in law and the measurability of the mul-tifunctions ( t, x ) → D + t + ,x V ( t, x ) to obtain ( p ⋆ ( t, x ) , q ⋆ ( t, x ) , Θ ( t, x )) ∈ D + t + ,x V ( t, x ) thatminimizes (4.3) by virtue of the celebrated Filippov’s Lemma. The rest parts we can get from[7] or [13]. Acknowledgments.
The author would like to thank the anonymous referee for thecareful reading of the manuscript and helpful suggestions.