New approach to optimal control of delayed stochastic Volterra integral equations

Ne w approach to optimal control of delayed stochastic V olterra inte gral equations Roméo K ouassi K onan * and Auguste Aman † UFR Mathématiques et Informatique, Uni versité Félix H. Boign y , Cocody , Abstract W e address the optimal control of stochastic V olterra integral equations with delay through the lens of Hida-Malliavin calculus. W e sho w that the corresponding adjoint processes satisfy an anticipated backward stochastic V olterra integral equation (ABSVIE), and, e xploiting this structure, we establish both necessary and sufficient stochastic maximum principles. Our re- sults provide a comprehensive and rigorous frame work for characterizing optimal controls in delayed stochastic systems. MSC :Primary: 60F05, 60H15; Secondary: 60J30 K eywords : Backward stochastic V olterra integral equations, time delayed generators, Hölder continuity condition. 1 Intr oduction Optimal control of stochastic systems has been a central topic in applied mathematics, with ap- plications spanning finance, engineering, and biology . In particular , stochastic V olterra integral equations (SVIEs) with delay arise naturally in models where the e volution of the system depends not only on its current state b ut also on its past history . Delays and memory ef fects introduce signif- icant mathematical challenges, making the analysis and control of such systems a rich and activ e area of research. T raditional approaches for controlling stochastic differential equations (SDEs) often rely on backward stochastic dif ferential equations (BSDEs) and the classical stochastic max- imum principle. Ho wev er , for SVIEs with delay , the non-Markovian nature of the system pre vents a straightforward application of these classical tools. Recent de velopments have highlighted the role of anticipated BSDEs, which naturally accommodate the forward-looking dependence on fu- ture v alues of the adjoint processes, as well as Hida-Mallia vin calculus, which provides po werful techniques for handling anticipati ve stochastic integrals. In this work, we propose a novel approach * romeokouadiokonan071@gmail.com † aman.auguste@ufhb .edu.ci/augusteaman5@yahoo.fr , corresponding author to the optimal control of stochastic V olterra integral equations with delay . By exploiting the dual- ity between stochastic delay dif ferential equations (SDDEs) and anticipated backward stochastic dif ferential equations (ABSDES), we deriv e a rigorous frame work to characterize optimal con- trols. W e establish both necessary and sufficient conditions for optimality through stochastic max- imum principles, providing theoretical tools that extend the classical results to the delayed, non- Marko vian setting. Our approach opens new a venues for the analysis and computation of optimal strategies in systems with memory and delay effects. More precisely , in this paper we consider a controlled stochastic V olterra-type integral equation with delay of the form:                  X u ( t ) = x 0 ( t ) + Z t 0 b ( t , s , X u ( s − δ ) , u ( s )) d s + Z t 0 σ ( t , s , X u ( s − δ ) , u ( s )) d B ( s ) , t ∈ [ 0 , T ] , X u ( t ) = x 0 ( t ) , t ∈ [ − δ , 0 ] , (1.1) where x 0 , b and σ are some giv en function and u denotes the control process, assumed to take its values in a giv en set U and B is a Bro wnian motion defined on a filtered probability space  Ω , F , { F t } t ⩾ 0 , P  satisfying the usual conditions. For a given function f and g , let us consider the performance functional J defined by: for all u ∈ U , J ( x 0 , u ) = E  Z T 0 f ( t , X u ( t − δ ) , u ( t )) d t + g ( X u ( T )) | X u ( 0 ) = x 0 ( 0 )  . The main moti v ation of this paper is to establish an existence of a control u ∗ ∈ U called an optimal control that maximizes the performance functional J such that φ ( x 0 ) = J ( x 0 , u ∗ ) = sup u ∈ U J ( x 0 , u ) . (1.2) Se veral variants of this problem hav e already been in vestigated in the frame work of optimal control of stochastic V olterra inte gral equations (SVIEs), notably by Y ong [27, 28] and by Agram et al. [2, 3]. Ho wev er , to the best of our knowledge, despite these contributions, the corresponding approach has not yet been extended to delayed systems in the conte xt of stochastic V olterra differential equations. One may also mention the works of Bernt Øksendal, Agnès Sulem, and T usheng Zhang [8], who analyzed optimal control with delay in the setting of stochastic dif ferential equations with memory , dri ven by both a Bro wnian motion and a Poisson random measure. Furthermore, in the contributions of Nacira Agram, Bernt Øksendal, and Samia Y akhlef [3], the control problem is reformulated within the frame work of stochastic V olterra equations. Our approach in this paper dif fers from the aforementioned works. In particular , the presence of the terms X ( s − δ ) renders the problem non-Markovian, thereby precluding the use of a finite- dimensional dynamic programming approach. Nev ertheless, we demonstrate that it is still possible to deri ve a Pontryagin-Bismut-Bensoussan type stochastic maximum principle. 2 The paper is organized as follo ws. Section 2 introduces the formulation of the control problem and presents some preliminary results. In Section 3, we deriv e the Hamiltonian function together with the corresponding adjoint equation associated with our control problem. Finally , Sections 4 and 5 are dev oted respectively to the establishment of a sufficient and a necessary stochastic maximum principle. 2 F ormulation of problem Let ( Ω , F , { F t } t ≥ 0 , P ) be a filtered probability space equipped with a Brownian motion ( B ( t )) t ∈ [ 0 , T ] satisfying the usual conditions and U designed an admissible control set. For u ∈ U , we recall the controlled state ( X u ( t )) t ≥ 0 , described by a stochastic V olterra integral equation with delay                  X u ( t ) = x 0 ( t ) + Z t 0 b ( t , s , X u ( s − δ ) , u ( s )) d s + Z t 0 σ ( t , s , X u ( s − δ ) , u ( s )) d B ( s ) , t ∈ [ 0 , T ] , X u ( t ) = x 0 ( t ) , t ∈ [ − δ , 0 ] , (2.1) and the cost functional J ( x 0 , u ) = E  Z T 0 f ( t , X u ( t − δ ) , u ( t )) d t + g ( X u ( T )) | ( X u s ) s ∈ [ − δ , 0 ] = x 0  . (2.2) Throughout this work, we will use the follo wing spaces: • S 2 ( R ) is the set R -valued { F t } t ≥ 0 -adapted continuous process ( ϕ ( t )) t ≥ 0 such that E sup 0 ≤ t ≤ T | ϕ ( t ) | 2 ! < + ∞ . • M 2 ( R ) is the set R -valued { F t } t ≥ 0 -adapted continuous process ( ϕ ( t )) t ≥ 0 such that E  Z T 0 | ϕ ( t ) | 2 d t  < + ∞ . T o conclude this section, we recall some notions and results related to the Hida-Malliavin deriv a- ti ve, which, according to our approach, is very important for deriving the adjoint equation and the related results. Indeed, let suppose u ∗ be a optimal control and denote u ε = u ∗ + ε u , for u a gi ven admissible control. W e consider the process Y defined by Y ( t ) = d X u ε ( t ) d ε | ε = 0 , 3 where X u ε is the solution of (2.1) with control u ε . Unlike the case of classical stochastic differential equations (SDEs), when the state equation is driv en by SVIE (2.1), it is not possible to directly isolate the quantity Y . T o do it, like in [3], we need a the help of Hida-Malliavin deriv ativ e. Let briefly gi ve notion some information of the notion. Let first consider simple random v ariables of the form F = f  Z T 0 h 1 ( t ) d B ( t ) , · · · , Z T 0 h n ( t ) d B ( t )  , where f ∈ C ∞ ( R n ) and h i ∈ L 2 ([ 0 , T ]) for i = 1 , · · · , n . The Malliavin deri vati ve D t F is then defined by: D t F = n ∑ i = 1 ∂ f ∂ x i  Z T 0 h 1 ( t ) d B ( t ) , · · · , Z T 0 h n ( t ) d B ( t )  h i ( t ) , t ∈ [ 0 , T ] . Thus, D t F is a stochastic process (in t ), representing the sensitivity of F to a small perturbation of the noise B at time t . he operator D is closable in L 2 ( Ω ) which allows one to define a stochastic Sobole v space: D 1 , 2 = { F ∈ L 2 ( Ω ) , DF ∈ L 2 ([ 0 , T ] × Ω ) } . One of the fundamental results is the Clark-Ocone duality formula Proposition 2.1. (Generalized Clark–Ocone F ormula [1]) F or all F ∈ L 2 ( F T , P ) , we have: F = E [ F ] + Z T 0 E [ D t F | F t ] d B ( t ) . Moreov er , the following generalized duality formula holds for Bro wnian motion. Proposition 2.2. (Generalized Duality F ormula for the Brownian motion B ) F ix s ∈ [ 0 , T ] . If t 7→ ϕ ( t , s , ω ) ∈ L 2 ( ε × P ) is an F -adapted pr ocess with E  Z T t ϕ 2 ( t , s ) d t  < ∞ , and F ∈ L 2 ( F T , P ) . Then we have E  F Z T 0 ϕ ( t , s ) d B ( t )  = E  Z T 0 E [ D t F | F t ] ϕ ( t , s ) d t  . (2.3) Pr oof. As observed by Agram and Øksendal [2], for a fixed s ∈ [ 0 , T ] , using Proposition 2.1 to- gether with Itô’ s isometry , we obtain E  Z T t ϕ ( s , t ) d B ( t )  F  = E  E [ F ] + Z T t E [ D t F | F t ] d B ( t )   Z T t ϕ ( s , t ) d B ( t )  = E  Z T t E [ D t F | F t ] ϕ ( s , t ) d t  . 4 Theorem 2.1. (Representation Theorem for advanced BSVIEs) Assume that the g enerator φ : [ 0 , T ] 2 × R × R → R such that φ ( ., s ., . ) is ( F s ) s ≥ 0 -adapted. Suppose that ther e is the couple of pr ocess ( p ( t ) , q ( t , . )) solution of the bac kward stochastic V olterra inte gral equation (BSVIE) p ( t ) = Φ ( t ) + Z T t φ ( t , s , p ( s + δ ) , q ( t , s )) d s − Z T t q ( t , s ) d B ( s ) , t ∈ [ 0 , T ] . Then, for all 0 ≤ t < s ≤ T , we have q ( t , s ) = E [ D s p ( t ) | F s ] . (2.4) 3 T ime-advanced BSVIE f or adjoint equations In this section, we focus on deri ving the adjoint equation associated with the stochastic V olterra integral equation (SVIE) (2.1) by applying the stochastic maximum principle. ( A1 ) For all x , u , the processes b ( ., s , x , u ) , σ ( ., s , x , u ) and f ( ., s , x , u , p , q ) are ( F s ) s ≥ 0 -adapted and twice continuously differentiable C 2 b with respect to t , x and continuously dif ferentiable C 1 b with respect to u . ( A2 ) The function g is F T -measurable and C 1 b class. ( A3 ) The function t 7→ q ( t , . ) is of class C 1 and E " Z T 0 Z T 0  ∂ q ( t , s ) ∂ t  2 d s d t # < + ∞ . Theorem 3.1. (Stochastic Maximum Principle: Adjoint Equation) Assume ( A1 ) - ( A3 ) . Then ther e exists an adapted adjoint pr ocess ( p ( t ) , q ( ., t )) satisfying the fol- lowing advanced backwar d stochastic V olterra equation p ( t ) = ∂ g ∂ x  X ( T )  + Z T t µ ( s , X ( s ) , p ( s + δ ) , q ( t , s + δ )) d s − Z T t q ( t , s ) d B ( s ) (3.1) wher e µ ( t , X ( t ) , p ( t + δ ) , q ( t , t + δ )) = ∂ f ∂ x ( s + δ , X u ( s ) , u ( s + δ )) + ∂ b ∂ x ( s + δ , t + δ ) p ( s + δ ) + Pr oof. Le consider the s tochastic V olterra integral equation with delay of the form      X u ( t ) = Z t 0 b ( t , s , X u ( s − δ ) , u ( s )) d s + Z t 0 σ ( t , s , X u ( s − δ ) , u ( s )) d B ( s ) t ∈ [ 0 , T ] , X u ( t ) = x 0 ( t ) , t ∈ [ − δ , 0 ] , δ > 0 , 5 and recall the optimal problem φ ( x 0 ) = sup u ∈ U J ( x 0 , u ) , where the cost functional J is defined by (2.2). For sufficiently small ε > 0, let us define u ε ( t ) = u ∗ ( t ) + εβ ( t ) , where u ∗ and β are respectiv ely optimal and admissible control. Since ε > 0 is suf ficiently small, u ε belongs to U . Then, according to assumption ( A1 ) and ( A2 ) we hav e d J ( x 0 , u ε ) d ε | ε = 0 = lim ε → 0 J ( x 0 , u ε ) − J ( x 0 , u ) ε = lim ε → 0 E " Z T 0 f ( t , X u ε ( t − δ ) , u ε ( t )) − f ( t , X u ( t − δ ) , u ( t )) ε ! d t + g  X u ε ( T )  − g  X u ( T ) ε    ( X u s ) s ∈ [ − δ , 0 ] = x 0 # = E " Z T 0 lim ε → 0 f ( t , X u ε ( t − δ ) , u ε ( t )) − f ( t , X u ( t − δ ) , u ( t )) ε ! d t + lim ε → 0 g  X u ε ( T )  − g  X u ( T ) ε    ( X u s ) s ∈ [ − δ , 0 ] = x 0 # = E  Z T 0  ∂ f ( t , X u ( t − δ ) , u ( t )) ∂ x Y ( t − δ ) + ∂ f ( t , X u ( t − δ ) , u ( t )) ∂ u β ( t )  d t + ∂ g ∂ x ( X u ( T )) Y ( T )    ( X u s ) s ∈ [ − δ , 0 ] = x 0  , (3.2) where we recall that Y ( t ) = d X u ε ( t ) d ε | ε = 0 . On the the other hand, Let us suppose that an adjoint equation of (2.1) is the follo wing BSDE. p ( t ) = ∂ g ∂ x ( X u ( T )) + Z T t h ( s ) d s − Z T t q ( t , s ) d B ( s ) , t ∈ [ 0 , T ] , with it dif ferential form        d p ( t ) = −  h ( t ) + Z T t ∂ q ∂ t ( t , s ) d B ( s )  d t + q ( t , t ) d B ( t ) p ( T ) = ∂ g ∂ x ( X u ( T )) (3.3) In the sequel of this proof, our goal is to e xplicitly determine the e xpression of µ . For this instance, let us apply Itô’ s formula to p ( T ) Y ( T ) . W e hav e p ( T ) Y ( T ) = p ( 0 ) Y ( 0 ) + Z T 0 p ( t ) dY ( t ) + Z T 0 Y ( t ) d p ( t ) + Z T 0 d ⟨ p , Y ⟩ t . (3.4) 6 T aking expectation in (5.5) together with the fact that p ( T ) = ∂ g ∂ x ( X u ( T )) , we get E  ∂ g ∂ x ( X u ( T )) Y ( T )  = E  p ( 0 ) Y ( 0 ) + Z T 0 p ( t ) dY ( t ) + Z T 0 Y ( t ) d p ( t ) + Z T 0 d ⟨ p , Y ⟩ t  . (3.5) In vie w of it definition, we have Y ( t ) = Z t 0  ∂ b ∂ x ( t , s ) Y ( s − δ ) + ∂ b ∂ u ( t , s ) β ( s )  d s + Z t 0  ∂σ ∂ x ( t , s ) Y ( s − δ ) + ∂σ ∂ u ( t , s ) β ( s )  d B ( s ) and then dY ( t ) =  ∂ b ∂ x ( t , t ) Y ( t − δ ) + ∂ b ∂ u ( t , t ) β ( t ) + Z t 0 ∂ 2 b ∂ t ∂ x ( t , s ) Y ( s − δ ) d s + Z t 0 ∂ 2 b ∂ t ∂ u ( t , s ) β ( s ) d s + Z t 0 ∂ 2 σ ∂ t ∂ x ( t , s ) Y ( s − δ ) d B ( s ) + Z t 0 ∂ 2 σ ∂ t ∂ u ( t , s ) β ( s ) d B ( s )  d t +  ∂σ ∂ x ( t , t ) Y ( t − δ ) + ∂σ ∂ u ( t , t ) β ( t )  d B ( t ) , where φ ( t , s ) = φ ( t , s , X u ( s − δ ) , u ( s )) for φ = b , σ . Thus, we get E  Z T 0 p ( t ) dY ( t )  = E  Z T 0  ∂ b ∂ x ( t , t ) p ( t ) Y ( t − δ ) + ∂ b ∂ u ( t , t ) p ( t ) β ( t )  d t  + E  Z T 0  ∂σ ∂ x ( t , t ) p ( t ) Y ( t − δ ) + ∂σ ∂ u ( t , t ) p ( t ) β ( t )  d B ( t )  + E  Z T 0 p ( t )  Z T 0  ∂ 2 b ∂ t ∂ x ( t , s ) Y ( s − δ ) + ∂ 2 b ∂ t ∂ u ( t , s ) β ( s )  1 [ 0 , t ] ( s ) d s  d t  + E  Z T 0 p ( t )  Z T 0  ∂ 2 σ ∂ t ∂ x ( t , s ) Y ( s − δ ) + ∂ 2 b ∂ t ∂ u ( t , s ) β ( s )  1 [ 0 , t ] d B ( s )  d t  = E  Z T 0  ∂ b ∂ x ( t , t ) p ( t ) Y ( t − δ ) + ∂ b ∂ u ( t , t ) p ( t ) β ( t )  d t  + I 1 + I 2 + I 3 (3.6) In vie w of ( A1 ) - (( A2 ) , we hav e I 1 = 0 . (3.7) By using Fubini’ s theorem, we obtain I 2 = E  Z T 0  Z T 0 p ( t )  ∂ 2 b ∂ t ∂ x ( t , s ) Y ( s − δ ) + ∂ 2 b ∂ t ∂ u ( t , s ) β ( s )  1 [ s , + ∞ [ ( t ) d t  d s  . (3.8) 7 Using again Fubini’ s Theorem and in virtue of the duality formula (2.3), we obtain I 3 = Z T 0 E  p ( t ) Z T 0  ∂ 2 σ ∂ t ∂ x ( t , s ) Y ( s − δ ) + ∂ 2 σ ∂ t ∂ u ( t , s ) β ( s )  1 [ 0 , t ]( s ) d B ( s )  d t = Z T 0 E  Z T 0 E [ D s p ( t ) | F s ]  ∂ 2 σ ∂ t ∂ x ( t , s ) Y ( s − δ ) + ∂ 2 σ ∂ t ∂ u ( t , s ) β ( s )  1 [ 0 , t ]( s ) d s  d t = E  Z T 0  Z T 0 E [ D s p ( t ) | F s ]  ∂ 2 σ ∂ t ∂ x ( t , s ) Y ( s − δ ) + ∂ 2 σ ∂ t ∂ u ( t , s ) β ( s )  1 [ s , + ∞ ] ( t ) d t  d s  Next, according to (2.4) in Theorem 2.1, we ha ve I 3 = E  Z T 0  Z T 0 q ( t , s )  ∂ 2 σ ∂ t ∂ x ( t , s ) Y ( s − δ ) + ∂ 2 σ ∂ t ∂ u ( t , s ) β ( s )  1 [ s , + ∞ ] ( t ) d t  d s  . (3.9) Plugging (5.8)-(5.10) in (5.7) we obtain E  Z T 0 p ( t ) dY ( t )  = E  Z T 0  ∂ b ∂ x ( t , t ) p ( t ) Y ( t − δ ) + ∂ b ∂ u ( t , t ) p ( t ) β ( t )  d t + Z T 0  Z T 0 p ( t )  ∂ 2 b ∂ t ∂ x ( t , s ) Y ( s − δ ) + ∂ 2 b ∂ t ∂ u ( t , s ) β ( s )  1 [ s , + ∞ [ ( t ) d t  d s + Z T 0  Z T 0 q ( t , s )  ∂ 2 σ ∂ t ∂ x ( t , s ) Y ( s − δ ) + ∂ 2 σ ∂ t ∂ u ( t , s ) β ( s )  1 [ s , + ∞ ] ( t ) d t  d s  . (3.10) On the other hand, it follo ws from (3.3), (3.6) that E  Z T 0 d ⟨ p , Y ⟩ t  = E  Z T 0 q ( t , t )  ∂σ ∂ x ( t , t ) Y ( t − δ ) + ∂σ ∂ x ( t , t ) β ( t )  d t  (3.11) and E  Z T 0 Y ( t ) d p ( t )  = E  Z T 0 Y ( t )  − h ( t , X ( t ) , p ( t ) , q ( t , t )) − Z T t ∂ q ∂ t ( t , s ) d B ( s )  d t + Z T 0 Y ( t ) q ( t , t ) d B ( t )  = − E  Z T 0 Y ( t ) h ( t , X ( t ) , p ( t ) , q ( t , t )) dt  − E  Z T 0 Y ( t )  Z T t ∂ q ∂ t ( t , s ) d B ( s )  d t  + E  Z T 0 Y ( t ) q ( t , t ) d B ( t )  = − E  Z T 0 Y ( t ) h ( t , X ( t ) , p ( t ) , q ( t , t )) dt  . (3.12) 8 Indeed, in view of assumption ( A1 ) - ( A3 ) and stochastic Fubini Theorem, one can deri ve easily that E  Z T 0 Y ( t )  Z T t ∂ q ∂ t ( t , s ) d B ( s )  d t  = E  Z T 0  Z s 0 Y ( t ) ∂ q ∂ t ( t , s ) d t  d B ( s )  = 0 and E  Z T 0 Y ( t ) q ( t , t ) d B ( t )  = 0 . Thus, with (5.11), (5.12), (5.13) and (5.6) put together , we obtain E  ∂ g ∂ x ( X u ( T )) Y ( T )  = E  Z T 0  ∂ b ∂ x ( t , t ) p ( t ) Y ( t − δ ) + ∂ b ∂ u ( t , t ) p ( t ) β ( t )  d t + Z T 0 q ( t , t )  ∂σ ∂ x ( t , t ) Y ( t − δ ) + ∂σ ∂ x ( t , t ) β ( t )  d t + Z T 0  Z T 0 p ( t )  ∂ 2 b ∂ t ∂ x ( t , s ) Y ( s − δ ) + ∂ 2 b ∂ t ∂ u ( t , s ) β ( s )  1 [ s , + ∞ [ ( t ) d t  d s + Z T 0  Z T 0 q ( t , s )  ∂ 2 σ ∂ t ∂ x ( t , s ) Y ( s − δ ) + ∂ 2 σ ∂ t ∂ u ( t , s ) β ( s )  1 [ s , + ∞ ] ( t ) d t  d s − Z T 0 Y ( t ) h ( t , t , X ( t ) , p ( t ) , q ( t , s )) d t  9 Finally , we hav e d J ( x 0 , u ε ) d ε | ε = 0 = E  Z T 0  ∂ f ( t , X u ( t − δ ) , u ( t )) ∂ x Y ( t − δ ) + ∂ f ( t , X u ( t − δ ) , u ( t )) ∂ u β ( t )  d t + Z T 0  ∂ b ∂ x ( t , t ) p ( t ) Y ( t − δ ) + ∂ b ∂ u ( t , t ) p ( t ) β ( t )  d t + Z T 0 q ( t , t )  ∂σ ∂ x ( t , t ) Y ( t − δ ) + ∂σ ∂ x ( t , t ) β ( t )  d t + Z T 0  Z T 0 p ( t )  ∂ 2 b ∂ t ∂ x ( t , s ) Y ( s − δ ) + ∂ 2 b ∂ t ∂ u ( t , s ) β ( s )  1 [ s , + ∞ [ ( t ) d t  d s + Z T 0  Z T 0 q ( t , s )  ∂ 2 σ ∂ t ∂ x ( t , s ) Y ( s − δ ) + ∂ 2 σ ∂ t ∂ u ( t , s ) β ( s )  1 [ s , + ∞ ] ( t ) d t  d s + Z T 0 Y ( t ) d p ( t )  = E  Z T − δ 0  ∂ f ( t + δ , X u ( t ) , u ( t + δ )) ∂ x Y ( t ) + ∂ f ( t + δ , X u ( t ) , u ( t + δ )) ∂ u β ( t + δ )  d t Z T − δ 0  ∂ b ∂ x ( t + δ , t + δ ) p ( t + δ ) Y ( t ) + ∂ b ∂ u ( t + δ , t + δ ) p ( t + δ ) β ( t + δ )  d t + Z T − δ 0 q ( t + δ , t + δ )  ∂σ ∂ x ( t + δ , t + δ ) Y ( t ) + ∂σ ∂ x ( t + δ , t + δ ) β ( t + δ )  d t + Z T 0  Z T 0 p ( r )  ∂ 2 b ∂ r ∂ x ( r , s + δ ) Y ( s ) + ∂ 2 b ∂ r ∂ u ( r , s + δ ) β ( s + δ )  1 [ s + δ , + ∞ [ ( r ) d r  d s + Z T 0  Z T 0 q ( r , s + δ )  ∂ 2 σ ∂ r ∂ x ( r , s + δ ) Y ( s ) + ∂ 2 σ ∂ r ∂ u ( r , s + δ ) β ( s + δ )  1 [ s + δ , + ∞ [ ( r ) d r  d s − Z T 0 Y ( t ) h ( t ) d t  . Since adjoint process in order to eliminate the terms in volving Y and to express the v ariation solely in terms of the control perturbation, we need for all t ∈ [ 0 , T − δ ] , ∂ f ∂ x ( t + δ , X u ( t ) , u ( t + δ )) + ∂ b ∂ x ( t + δ , t + δ ) p ( t + δ ) + ∂σ ∂ x ( t + δ , t + δ ) q ( t + δ , t + δ ) + Z T t + δ  p ( s ) ∂ 2 b ∂ s ∂ x ( s , t + δ ) d s + Z T t + δ q ( s , t + δ ) ∂ 2 σ ∂ s ∂ x ( s , t + δ )  d s − h ( t ) = 0 so that h ( t ) =  ∂ f ∂ x ( t + δ , X u ( t ) , u ( t + δ )) + ∂ b ∂ x ( t + δ , t + δ ) p ( t + δ ) + ∂σ ∂ x ( t + δ , t + δ ) q ( t + δ , t + δ ) + Z T t + δ  p ( s ) ∂ 2 b ∂ s ∂ x ( s , t + δ ) d s + Z T t + δ q ( s , t + δ ) ∂ 2 σ ∂ s ∂ x ( s , t + δ )  d s  1 [ 0 , T − δ ] ( t ) . 10 Remark 3.2 . W e observe that equation ( ?? ) is an anticipated V olterra-type BSDE. This type of BSDE was first studied by Jiaqiang W en and Y ufeng Shi [24]. In particular , they established a result on existence and uniqueness under a global Lipschitz condition. 4 Sufficient Maximum Principle In this section, we formulate a stochastic maximum principle for stochastic V olterra integral sys- tems with delay under partial information. The information av ailable to the controller is modeled by a sub-filtration G t ⊆ F t , for all t ≥ 0. W e assume that the control set U ⊂ R is con vex. The set of admissible controls, denoted by A G , consists of all càdlàg processes taking v alues in U and adapted to the filtration ( G t ) t ≥ 0. For notational simplicity , we write X u ( t ) = X ( t ) . Theorem 4.1. Let ˆ u ∈ A G be an admissible contr ol, and denote by ˆ X ( t ) the corr esponding state pr ocess. Let  ˆ p ( t ) , ˆ q ( t , s )  be the associated adjoint pr ocesses, which ar e solutions of equations (3.1) . Assume that the following conditions hold: (i) The function x 7→ g ( x ) and the Hamiltonian ( x , u ) 7→ H ( t , x , u , ˆ p ( t ) , ˆ q ( ., t )) are concave for each t ∈ [ 0 , T ] , almost sur ely . (ii) Assumption ( A3 ) holds for all u ∈ A G . (iii) (Maximum condition) F or all t ∈ [ 0 , T ] , it holds that E  H  t , ˆ X ( t − δ ) , ˆ u ( t ) , ˆ p ( t ) , ˆ q ( · , t )    G t  = max v ∈ U E  H ( t , ˆ X ( t − δ ) , v , ˆ p ( t ) , ˆ q ( · , t ))   G t  . (4.1) Then, the contr ol ˆ u is optimal for the contr ol pr oblem (1.2) . Pr oof. Consider the state process ˆ X ( t ) associated with the admissible control ˆ u ∈ A G , defined by the stochastic V olterra integral equation with delay      ˆ X ( t ) = Z t 0 b ( t , s , ˆ X ( s − δ ) , ˆ u ( s )) d s + Z t 0 σ  t , s , ˆ X ( s − δ ) , ˆ u ( s )  d B ( s ) , t ∈ [ 0 , T ] , ˆ X ( t ) = x 0 ( t ) , t ∈ [ − δ , 0 ] , where δ > 0 is a giv en delay . Let u ∈ A G be any admissible control. Our objective is to pro ve that J ( x 0 , u ) ≤ J ( x 0 , ˆ u ) , which implies that ˆ u is an optimal control. According to (2.2), we hav e J ( x 0 , u ) − J ( x 0 , ˆ u ) = E  Z T 0  f  t , X ( t − δ ) , u ( t )  − f  t , ˆ X ( t − δ ) , ˆ u ( t )   , d t  E  g  X ( T )  − g  ˆ X ( T )  = I 1 + I 2 . 11 In vie w of expression and assumption of H , we ha ve I 1 = E  Z T 0  H ( t , X ( t − δ ) , u ( t )) − H ( t , ˆ X ( t − δ ) , ˆ u ( t )) − ˆ p ( t )( b ( t , t , X ( t − δ ) , u ( t )) − b ( t , t , ˆ X ( t − δ ) , ˆ u ( t )) − ˆ q ( t , t )( σ ( t , t , X ( t − δ ) , u ( t )) − σ ( t , t , ˆ X ( t − δ ) , ˆ u ( t ))) − Z T t ˆ p ( s )  ∂ b ∂ s ( s , t , X ( s − δ ) , u ( s )) − ∂ b ∂ s ( s , t , ˆ X ( s − δ ) , ˆ u ( s ))  d s − Z T t ˆ q ( s , t )  ∂σ ∂ s ( s , t , X ( s − δ ) , u ( s )) − ∂σ ∂ s ( s , t , ˆ X ( s − δ ) , ˆ u ( s ))  d s  d t  ≤ E  Z T 0  ∂ ˆ H ∂ x ( t )( X ( t − δ ) − ˆ X ( t − δ )) + ∂ ˆ H ∂ u ( t )( u ( t ) − ˆ u ( t )) − ˆ p ( t )( b ( t , t ) − ˆ b ( t , t )) − ˆ q ( t , t )( σ ( t , t ) − ˆ σ ( t , t )) − Z T t ˆ p ( s )( ∂ b ∂ s ( s , t ) − ∂ ˆ b ∂ s ( s , t )) d s − Z T t ˆ q ( s , t )( ∂σ ∂ s ( s , t ) − ∂ ˆ σ ∂ s ( s , t )) d s  d t  , Abbre viated notations: ∂ ˆ H ∂ x ( t ) : = ∂ H ∂ x ( t , ˆ X ( t − δ ) , ˆ u ( t ) , ˆ p ( t ) , ˆ q ( · , t )) , ∂ ˆ H ∂ u ( t ) : = ∂ H ∂ u ( t , ˆ X ( t − δ ) , ˆ u ( t ) , ˆ p ( t ) , ˆ q ( · , t )) , b ( s , t ) : = b ( s , t , X ( s − δ ) , u ( s )) , ˆ b ( s , t ) : = b ( s , t , ˆ X ( s − δ ) , ˆ u ( s )) . b ( t , t ) : = b ( t , t , X ( t − δ ) , u ( t )) , ˆ b ( t , t ) : = b ( t , t , ˆ X ( t − δ ) , ˆ u ( t )) . σ ( t , t ) : = σ ( t , t , X ( t − δ ) , u ( t )) , ˆ σ ( t , t ) : = σ ( t , t , ˆ X ( t − δ ) , ˆ u ( t )) . Using the concavity of the function g together with the terminal condition associated with the BSVIE, we deduce that I 2 ≤ E  ∂ g ∂ x  ˆ X ( T )  ·  ˆ X ( T ) − ˆ X ( T )   ≤ E  ˆ p ( T ) ·  X ( T ) − ˆ X ( T )  W e apply Itô’ s formula to ˆ p ( t )  X ( t ) − ˆ X ( t )  , we hav e ˆ p ( t )  X ( t ) − ˆ X ( t )  = ˆ p ( 0 )  X ( 0 ) − ˆ X ( 0 )  + Z T 0 ˆ p ( t ) d  X ( t ) − ˆ X ( t )  + Z T 0  X ( t ) − ˆ X ( t )  d ˆ p ( t ) + Z T 0 d ⟨ p ,  X − ˆ X  ⟩ t . (4.2) 12 By taking expectation in (4.2) and using that ˆ p ( T ) = ∂ g ∂ x ( ˆ X ( T )) , we obtain E  ∂ g ∂ x ( ˆ X ( T ))  X ( t ) − ˆ X ( t )   = E  ˆ p ( 0 )  X ( 0 ) − ˆ X ( 0 )  + Z T 0 ˆ p ( t ) d  X ( t ) − ˆ X ( t )  + Z T 0  X ( t ) − ˆ X ( t )  d ˆ p ( t ) + Z T 0 d ⟨ ˆ p ,  X − ˆ X  ⟩ t  . (4.3) Thus, we get E  Z T 0 ˆ p ( t ) d  X ( t ) − ˆ X ( t )   = E  Z T 0 ˆ p ( t )  b ( t , t ) − ˆ b ( t , t )  d t  + E  Z T 0 ˆ p ( t ) ( σ ( t , t ) − ˆ σ ( t , t )) d B ( t )  + E " Z T 0 ˆ p ( t ) Z t 0 ∂ b ∂ t ( t , s ) − ∂ ˆ b ∂ t ( t , s ) ! d s ! d t # + E  Z T 0 ˆ p ( t )  Z t 0  ∂σ ∂ t ( t , s ) − ∂ ˆ σ ∂ t ( t , s )  d B ( s )  d t = E  Z T 0 ˆ p ( t )  b ( t , t ) − ˆ b ( t , t )  d t  + r 1 + r 2 + r 3 (4.4) Under assumptions ( A1 ) – ( A2 ) , we ha ve r 1 = 0 . (4.5) By using Fubini’ s theorem, we obtain r 2 = E " Z T 0 Z T s ˆ p ( t ) ∂ b ∂ t ( t , s ) − ∂ ˆ b ∂ t ( t , s ) ! d t ! d s # = E " Z T 0 Z T t ˆ p ( s ) ∂ b ∂ s ( s , t ) − ∂ ˆ b ∂ s ( s , t ) ! d s ! d t # . Using Fubini’ s theorem again and the duality formula (2.3), we obtain r 3 = Z T 0 E  ˆ p ( t ) Z t 0  ∂σ ∂ t ( t , s ) − ∂ ˆ σ ∂ t ( t , s )  d B ( s )  d t = Z T 0 E  Z t 0 E [ D s ˆ p ( t ) | F s ]  ∂σ ∂ t ( t , s ) − ∂ ˆ σ ∂ t ( t , s )  d s  d t = E  Z T 0  Z T s E [ D s ˆ p ( t ) | F s ]  ∂σ ∂ t ( t , s ) − ∂ ˆ σ ∂ t ( t , s )  d t  d s  = E  Z T 0  Z T t E [ D t p ( s ) | F t ]  ∂σ ∂ s ( s , t ) − ∂ ˆ σ ∂ s ( s , t )  d s  d t  . 13 Next, according to (2.4) in Theorem 2.1, we ha ve r 3 = E  Z T 0  Z T t ˆ q ( s , t )  ∂σ ∂ s ( s , t ) − ∂ ˆ σ ∂ s ( s , t )  d s  d t  . (4.6) On the other hand, we hav e E  Z T 0 d ⟨ ˆ p , X − ˆ X ⟩ t  = E  Z T 0 ˆ q ( t , t ) ( σ ( t , t ) − ˆ σ ( t , t )) d t  (4.7) and E  Z T 0  X ( t ) − ˆ X ( t )  d ˆ p ( t )  = E  Z T 0  X ( t ) − ˆ X ( t )   − ∂ ˆ H ∂ x ( t + δ ) 1 [ 0 , T − δ ] − Z T t ∂ ˆ q ∂ t ( t , s ) d B ( s )  d t + Z T 0  X ( t ) − ˆ X ( t )  ˆ q ( t , t ) d B ( t )  = − E  Z T 0  X ( t ) − ˆ X ( t )  ∂ ˆ H ∂ x ( t + δ ) 1 [ 0 , T − δ ] d t  − E  Z T 0  X ( t ) − ˆ X ( t )   Z T t ∂ ˆ q ∂ t ( t , s ) d B ( s )  d t  + E  Z T 0  X ( t ) − ˆ X ( t )  ˆ q ( t , t ) d B ( t )  = − E  Z T 0  X ( t ) − ˆ X ( t )  ∂ ˆ H ∂ x ( t + δ ) 1 [ 0 , T − δ ] d t  (4.8) Indeed, in view of assumption ( A1 ) - ( A3 ) and stochastic Fubini Theorem, one can deri ve easily that E  Z T 0  X ( t ) − ˆ X ( t )   Z T t ∂ ˆ q ∂ t ( t , s ) d B ( s )  d t  = E  Z T 0  Z s 0  X ( t ) − ˆ X ( t )  ∂ ˆ q ∂ t ( t , s ) d t  d B ( s )  = 0 and E  Z T 0  X ( t ) − ˆ X ( t )  ˆ q ( t , t ) d B ( t )  = 0 . Finally , we hav e I 2 ≤ E " Z T 0 ( ˆ p ( t )  b ( t , t ) − ˆ b ( t , t )  + ˆ q ( t , t ) ( σ ( t , t ) − ˆ σ ( t , t )) + Z T t ˆ p ( s ) ∂ b ∂ s ( s , t ) − ∂ ˆ b ∂ s ( s , t ) ! d s + Z T t ˆ q ( s , t )  ∂σ ∂ s ( s , t ) − ∂ ˆ σ ∂ s ( s , t )  d s − ∂ ˆ H ∂ x ( t + δ ) 1 [ 0 , T − δ ] ( t )  X ( t ) − ˆ X ( t )  ) d t # . (4.9) 14 By combining the results of inequalities ( ?? ) and (4.9), we obtain J ( x 0 , u ) − J ( x 0 , ˆ u ) ≤ E " Z T 0  ∂ ˆ H ∂ x ( t )  X ( t − δ ) − ˆ X ( t − δ )  + ∂ ˆ H ∂ u ( t )  u ( t ) − ˆ u ( t )  − ˆ p ( t )  b ( t , t ) − ˆ b ( t , t )  − ˆ q ( t , t ) ( σ ( t , t ) − ˆ σ ( t , t )) − Z T t ˆ p ( s )  ∂ b ∂ s ( s , t ) − ∂ ˆ b ∂ s ( s , t )  d s − Z T t ˆ q ( s , t )  ∂σ ∂ s ( s , t ) − ∂ ˆ σ ∂ s ( s , t )  d s + ˆ p ( t )  b ( t , t ) − ˆ b ( t , t )  + ˆ q ( t , t ) ( σ ( t , t ) − ˆ σ ( t , t )) + Z T t ˆ p ( s ) ∂ b ∂ s ( s , t ) − ∂ ˆ b ∂ s ( s , t ) ! d s + Z T t ˆ q ( s , t )  ∂σ ∂ s ( s , t ) − ∂ ˆ σ ∂ s ( s , t )  d s − ∂ ˆ H ∂ x ( t + δ ) 1 [ 0 , T − δ ] ( t )  X ( t ) − ˆ X ( t )  ) d t # Thus, J ( x 0 , u ) − J ( x 0 , ˆ u ) ≤ E " Z T 0 ∂ ˆ H ∂ x ( t )( X ( t − δ ) − ˆ X ( t − δ )) − Z T 0 ∂ ˆ H ∂ x ( t + δ ) 1 [ 0 , T − δ ] ( X ( t ) − ˆ X ( t )) d t + Z T 0 ∂ ˆ H ∂ u ( t )( u ( t ) − ˆ u ( t )) d t # ≤ E " Z T − δ 0 ∂ ˆ H ∂ x ( t + δ )( X ( t ) − ˆ X ( t )) − Z T 0 ∂ ˆ H ∂ x ( t + δ ) 1 [ 0 , T − δ ] ( X ( t ) − ˆ X ( t )) d t + Z T 0 ∂ ˆ H ∂ u ( t )( u ( t ) − ˆ u ( t )) d t # ≤ E " Z T 0  ∂ ˆ H ∂ x ( t + δ ) 1 [ 0 , T − δ ] ( t ) − ∂ ˆ H ∂ x ( t + δ ) 1 [ 0 , T − δ ]  ( X ( t ) − ˆ X ( t )) d t + Z T 0 ∂ ˆ H ∂ u ( t )( u ( t ) − ˆ u ( t )) d t # . ≤ E  Z T 0 ∂ ˆ H ∂ u ( t )( u ( t ) − ˆ u ( t )) d t  Since u and ˆ u are ε t -adapted and ˆ u maximizes the conditional Hamiltonian, we hav e E  ∂ ˆ H ∂ u ( t ) | ε t  = 0 , which implies E " E  ∂ ˆ H ∂ u ( t ) | ε t  ( u ( t ) − ˆ u ( t )) # = 0 . 15 Consequently , we hav e J ( x 0 , u ) − J ( x 0 , ˆ u ) ≤ E  Z T 0 ∂ ˆ H ∂ u ( t )  u ( t ) − ˆ u ( t )  d t  ≤ E  Z T 0 E  ∂ ˆ H ∂ u ( t )     ε t   u ( t ) − ˆ u ( t )  d t  ≤ 0 This prov es that ˆ u is an optimal control. 5 A necessary maximum principle One of the limitations of the sufficient maximum principle presented in Section 4 lies in the con- cavity condition, which is not always satisfied in practical cases. In this section, we establish a complementary result, going in the rev erse direction. More specifically , we will sho w that being a directional critical point for the performance functional J ( x 0 , u ) is equiv alent to being a critical point of the conditional Hamiltonian. For this, we introduce the follo wing assumptions (A1) For all u ∈ A ε and all bounded β ∈ A ε , there exists λ > 0 such that u + εβ ∈ A ε for all ε ∈ ] − λ , λ [ . (A2) For all t ∈ [ 0 , T ] and all bounded ε t -measurable random variables α = α ( t ) , let h ∈ [ T − t , T ] and the control process β ( t ) defined by β ( s ) : = α 1 [ t , t + h ] ( s ) , s ∈ [ 0 , T ] . (5.1) belongs to A ε . (A3) For e very bounded β ∈ A ε , the directional deri vati ve process Y ( t ) = d X u + εβ ( t ) d ε | ε = 0 ., (5.2) exists and belongs to L 2 ( ε × P ) . It follo ws from (1.1) that dY ( t ) = ∂ b ∂ x ( t , t ) Y ( t − δ ) + ∂ b ∂ u ( t , t ) β ( t ) + + Z t 0 ∂ 2 b ∂ t ∂ x ( t , s ) Y ( s − δ ) d s + Z t 0 ∂ 2 b ∂ t ∂ u ( t , s ) β ( s ) d s + Z t 0 ∂ 2 σ ∂ t ∂ x ( t , s ) Y ( s − δ ) d B ( s ) + Z t 0 ∂ 2 σ ∂ t ∂ u ( t , s ) β ( s ) d B ( s ) # d t + ∂σ ∂ x ( t , t ) Y ( t − δ ) d B ( t ) + ∂σ ∂ u ( t , t ) β ( t ) d B ( t ) (5.3) 16 And we use the fact that Y ( t − δ ) = d X u + εβ ( t − δ ) d ε | ε = 0 , W e are now ready to state the result Theorem 5.1 (Necessary Maximum Principle) . Let ˆ u ∈ A ε with the associated state solution ˆ X ( t ) of equation (1.1) , and the adjoint pr ocesses ˆ p ( t ) , ˆ q ( · , t ) being the solution of equations (3.1) , as well as the derivative pr ocess ˆ Y ( t ) defined in (5.2). Assume that Assumption ( A3 ) holds. Then the following statements ar e equivalent. (i) F or every bounded β ∈ A ε , we have lim ε → 0 J ( x 0 , ˆ u + εβ ) − J ( x 0 , ˆ u ) ε = d J ( x 0 , ˆ u + εβ ) d ε | ε = 0 = 0 (ii) F or all t ∈ [ 0 , T ] , we have E  ∂ H ∂ u  t , ˆ X ( t − δ ) , u , ˆ p ( t ) , ˆ q ( · , t )      ε t  u = ˆ u ( t ) = 0 p.s. Pr oof. T o simplify the notations, we write ˆ u = u , ˆ X = X , ˆ p = p , and ˆ q = q henceforth. Assume that statement (i) is true. Consider d J ( x 0 , u + εβ ) d ε | ε = 0 = E " Z T 0 ∂ f ∂ x ( t ) Y ( t − δ ) + ∂ f ∂ u ( t ) β ( t ) ! d t + ∂ g ∂ x  X ( T )  Y ( T ) # Using the definition of H , we have d J ( x 0 , u + εβ ) d ε | ε = 0 = E " Z T 0 (  ∂ H ∂ x ( t ) − ∂ b ∂ x ( t , t ) p ( t ) − ∂σ ∂ x ( t , t ) q ( t , t ) − Z T t p ( s ) ∂ 2 b ∂ s ∂ x ( s , t ) d s − Z T t q ( s , t ) ∂ 2 σ ∂ s ∂ x ( s , t ) d s  Y ( t − δ ) + ∂ f ∂ u ( t ) β ( t ) ) d t + ∂ g ∂ x ( X ( T )) Y ( T ) # (5.4) Applying Itô’ s formula to p ( t ) · Y ( t ) , we get p ( T ) Y ( T ) = p ( 0 ) Y ( 0 ) + Z T 0 p ( t ) dY ( t ) + Z T 0 Y ( t ) d p ( t ) + Z T 0 d ⟨ p , Y ⟩ t . (5.5) T aking expectations in (5.5) and using that p ( T ) = ∂ g ∂ x ( X ( T )) , we obtain E  ∂ g ∂ x ( X ( T )) Y ( T )  = E  p ( 0 ) Y ( 0 ) + Z T 0 p ( t ) dY ( t ) + Z T 0 Y ( t ) d p ( t ) + Z T 0 d ⟨ p , Y ⟩ t  . (5.6) 17 Thus, we get E  Z T 0 p ( t ) dY ( t )  = E  Z T 0  ∂ b ∂ x ( t , t ) p ( t ) Y ( t − δ ) + ∂ b ∂ u ( t , t ) p ( t ) β ( t )  d t  + E  Z T 0  ∂σ ∂ x ( t , t ) p ( t ) Y ( t − δ ) + ∂σ ∂ u ( t , t ) p ( t ) β ( t )  d B ( t )  + E  Z T 0 p ( t )  Z t 0  ∂ 2 b ∂ t ∂ x ( t , s ) Y ( s − δ ) + ∂ 2 b ∂ t ∂ u ( t , s ) β ( s )  d s  d t  + E  Z T 0 p ( t )  Z t 0  ∂ 2 σ ∂ t ∂ x ( t , s ) Y ( s − δ ) + ∂ 2 b ∂ t ∂ u ( t , s ) β ( s )  d B ( s )  d t  = E  Z T 0  ∂ b ∂ x ( t , t ) p ( t ) Y ( t − δ ) + ∂ b ∂ u ( t , t ) p ( t ) β ( t )  d t  + v 1 + v 2 + v 3 (5.7) Under assumptions ( A1 ) – ( A2 ) , we ha ve v 1 = 0 . (5.8) By using Fubini’ s theorem, we obtain v 2 = E  Z T 0  Z T s p ( t )  ∂ 2 b ∂ t ∂ x ( t , s ) Y ( s − δ ) + ∂ 2 b ∂ t ∂ u ( t , s ) β ( s )  d t  d s  = E  Z T 0  Z T t p ( s )  ∂ 2 b ∂ s ∂ x ( s , t ) Y ( t − δ ) + ∂ 2 b ∂ s ∂ u ( s , t ) β ( t )  d s  d t  . (5.9) Using again Fubini’ s Theorem and in virtue of the duality formula (2.3), we obtain v 3 = Z T 0 E  p ( t ) Z t 0  ∂ 2 σ ∂ t ∂ x ( t , s ) Y ( s − δ ) + ∂ 2 σ ∂ t ∂ u ( t , s ) β ( s )  d B ( s )  d t = Z T 0 E  Z t 0 E [ D s p ( t ) | F s ]  ∂ 2 σ ∂ t ∂ x ( t , s ) Y ( s − δ ) + ∂ 2 σ ∂ t ∂ u ( t , s ) β ( s )  d s  d t = E  Z T 0  Z T s E [ D s p ( t ) | F s ]  ∂ 2 σ ∂ t ∂ x ( t , s ) Y ( s − δ ) + ∂ 2 σ ∂ t ∂ u ( t , s ) β ( s )  d t  d s  = E  Z T 0  Z T t E [ D t p ( s ) | F t ]  ∂ 2 σ ∂ s ∂ x ( s , t ) Y ( t − δ ) + ∂ 2 σ ∂ s ∂ u ( s , t ) β ( t )  d s  d t  . Next, according to (2.4) in Theorem 2.1, we ha ve v 3 = E  Z T 0  Z T t q ( s , t )  ∂ 2 σ ∂ s ∂ x ( s , t ) Y ( t − δ ) + ∂ 2 σ ∂ s ∂ u ( s , t ) β ( t )  d s  d t  . (5.10) 18 Substituting (5.8)–(5.10) into (5.7), we obtain E  Z T 0 p ( t ) dY ( t )  = E  Z T 0  ∂ b ∂ x ( t , t ) p ( t ) Y ( t − δ ) + ∂ b ∂ u ( t , t ) p ( t ) β ( t )  d t + Z T 0  Z T t p ( s )  ∂ 2 b ∂ s ∂ x ( s , t ) Y ( t − δ ) + ∂ 2 b ∂ s ∂ u ( s , t ) β ( t )  d s  d t + Z T 0  Z T t q ( s , t )  ∂ 2 σ ∂ s ∂ x ( s , t ) Y ( t − δ ) + ∂ 2 σ ∂ s ∂ u ( s , t ) β ( t )  d s  d t  . (5.11) On the other hand, we hav e E  Z T 0 d ⟨ p , Y ⟩ t  = E  Z T 0 q ( t , t )  ∂σ ∂ x ( t , t ) Y ( t − δ ) + ∂σ ∂ x ( t , t ) β ( t )  d t  (5.12) and E  Z T 0 Y ( t ) d p ( t )  = E  Z T 0 Y ( t )  − ∂ H ∂ x ( t + δ ) 1 [ 0 , T − δ ] ( t ) − Z T t ∂ q ∂ t ( t , s ) d B ( s )  d t  + E  Z T 0 Y ( t ) q ( t , t ) d B ( t )  = − E  Z T 0 Y ( t ) ∂ H ∂ x ( t + δ ) 1 [ 0 , T − δ ] ( t ) d t  − E  Z T 0 Y ( t )  Z T t ∂ q ∂ t ( t , s ) d B ( s )  d t  + E  Z T 0 Y ( t ) q ( t , t ) d B ( t )  = − E  Z T 0 Y ( t ) ∂ H ∂ x ( t + δ ) 1 [ 0 , T − δ ] ( t ) d t  . (5.13) Indeed, in vie w of assumptions ( A1 ) – ( A3 ) and the stochastic Fubini theorem, one easily obtains E  Z T 0 Y ( t )  Z T t ∂ q ∂ t ( t , s ) d B ( s )  d t  = E  Z T 0  Z s 0 Y ( t ) ∂ q ∂ t ( t , s ) d t  d B ( s )  = 0 and E  Z T 0 Y ( t ) q ( t , t ) d B ( t )  = 0 . 19 Finally ,with (5.11), (5.12), (5.13) and (5.6) put together , we obtain E  ∂ g ∂ x ( X ( T )) Y ( T )  = E " Z T 0 ∂ b ∂ x ( t , t ) p ( t ) + Z T t  ∂ 2 b ∂ s ∂ x ( s , t ) p ( s ) + ∂ 2 σ ∂ s ∂ x ( s , t ) q ( s , t )  d s ! Y ( t − δ ) d t + Z T 0 ∂ b ∂ u ( t , t ) p ( t ) + Z T t  ∂ 2 b ∂ s ∂ u ( s , t ) p ( s ) + ∂ 2 σ ∂ s ∂ u ( s , t ) q ( s , t )  d s ! β ( t ) d t − Z T 0 Y ( t ) ∂ H ∂ x ( t + δ ) 1 [ 0 , T − δ ] ( t ) d t + Z T 0  ∂σ ∂ x ( t , t ) Y ( t − δ ) + ∂σ ∂ u ( t , t ) β ( t )  q ( t , t ) d t # . (5.14) By substituting the expression of (5.14) in (5.4), we get d J ( x 0 , u + εβ ) d ε | ε = 0 = E " Z T 0 Y ( t − δ ) ∂ H ∂ x ( t ) d t − Z T 0 Y ( t ) ∂ H ∂ x ( t + δ ) 1 [ 0 , T − δ ] ( t ) d t + Z T 0 ∂ H ∂ u ( t ) β ( t ) d t # = E " Z T − δ 0 Y ( t ) ∂ H ∂ x ( t + δ ) d t − Z T 0 Y ( t ) ∂ H ∂ x ( t + δ ) 1 [ 0 , T − δ ] ( t ) d t + Z T 0 ∂ H ∂ u ( t ) β ( t ) d t # = E " Z T 0  Y ( t ) ∂ H ∂ x ( t + δ ) 1 [ 0 , T − δ ] ( t ) − Y ( t ) ∂ H ∂ x ( t + δ ) 1 [ 0 , T − δ ] ( t )  d t + Z T 0 ∂ H ∂ u ( t ) β ( t ) d t # = E " Z T 0 ∂ H ∂ u ( t ) β ( t ) d t # Using the definition β , we obtain d J ( x 0 , u + εβ ) d ε | ε = 0 = E " Z T 0 ∂ H ∂ u ( s ) β ( s ) d s # = E " Z t + h t ∂ H ∂ u ( s ) d s α # (5.15) No w suppose that d J ( x 0 , u + εβ ) d ε | ε = 0 = 0 . (5.16) Dif ferentiating the right-hand side of (5.15) at h = 0, we get d d h E  α Z t + h t ∂ H ∂ u ( s ) d s      h = 0 = E  α ∂ H ∂ u ( t )  . 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