Stochastic Maximum Principle for Hilbert Space Valued Forward-Backward Doubly SDEs with Poisson Jumps

Abstract

Abstract. In this paper we study the stochastic maximum principle for a control problem in infinite dimensions. This problem is governed by a fully coupled forward-backward doubly stochastic differential equation (FBDSDE) driven by two cylindrical Wiener processes on separable Hilbert spaces and a Poisson random measure. We allow the control variable to enter in all coefficients appearing in this system. Existence and uniqueness of the solutions of FBDSDEs and an extended martingale representation theorem are provided as well.

Existence and uniqueness of the solutions of forward-backward doubly stochastic differential equations with Poisson jumps

“…Moreover, this work is crucial for addressing stochastic control problems associated with FBDSDEJ in infinite dimensions using the maximum principle approach. Further insights on this topic can be found in our previous works, namely [2] and [3]. Recently, our research in [5] highlighted the significant role of this paper in deriving strong solutions for FBDSDEs of McKean-Vlasov type.…”

“…From Theorems (5.3, 5.4) in [2], we know that (3.2) has a unique solution (y, Y, z, Z) in M 2 ([0, T ] , H 2 ). For more details, we refer the readers to the arguments presented in [2,3].…”

“…However, when working more generally over infinite dimensions, these cases necessarily change. To the best of our knowledge, apart from Theorem 2 in our previous work [2], which lacked a proof, no other work in the literature addresses such systems of FBDSDEs in infinite dimensions.…”

Existence and uniqueness of the solutions of forward-backward doubly stochastic differential equations with Poisson jumps

“…In fact, this work furnishes now a solid ground for studying such stochastic control problems governed by FBDSDEJ as in [2] in terms of the maximum principle approach. One can see also [1]. Applications of FBDSDEs with jumps to semilinear stochastic PDEs can be developed to give a probabilistic representation for the solution of a semilinear stochastic partial differential-integral equation in parallel to [13] and [4].…”

“…One can simply apply a generalized martingale representation theorem (as in Al-Hussien and Gherbal [1]) to get an explicit formula for this unique solution (y, z) since all integrals here do not depend on y or z.…”