Necessary and Sufficient Optimality Conditions for Relaxed and Strict Control of Forward-Backward Doubly SDEs with Jumps Under Full and Partial Information

Al-Hussein, AbdulRahman; Gherbal, Boulakhras

doi:10.1007/s11424-020-9013-3

Cited by 4 publications

(6 citation statements)

References 36 publications

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“…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.…”

Section: Introductionmentioning

confidence: 63%

“…In the following, we present our main results. For the proof of this theorem, one can closely follow the proof of Theorem 2.3 in [3], with minor modifications. Thus, we omit it in order to avoid increasing the size of this paper.…”

Section: Resultsmentioning

confidence: 99%

“…As we saw in Section 3, the condition 0 < γ < 1 2 crucial to guarantee the existence of solutions to the adjoint equations of MV-FBDSDE (4.2) in Theorem 4.2 below, which is one of the main theorems of this subsection. For additional clarification and similar discussions, refer to Remark 4.2 (i) in [5].…”

Section: Moreover Noting Thatmentioning

confidence: 99%

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

Al-Hussein

Gherbal

2020

Random Operators and Stochastic Equations

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The paper addresses a system of nonlinear fully coupled forward-backward doubly stochastic differential equations with Poisson jumps. These equations are allowed to live in Euclidean spaces of different dimensions, and the system is Markovian in the sense that the terminal value of the backward equation depends on the terminal value of the solution of the forward one. Under some monotonicity conditions we establish the existence and uniqueness of strong solutions of such equations by using a continuation method.

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Section: Introductionmentioning

confidence: 63%

Section: Resultsmentioning

confidence: 99%

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

Al-Hussein

Gherbal

2020

Random Operators and Stochastic Equations

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“…On the other hand, we are now able to apply our work here to stochastic maximum principle for FBDSDEJ. 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].…”

Section: Introductionmentioning

confidence: 92%

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

Al-Hussein¹,

Gherbal²

2018

Preprint

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The aim of this paper is to establish the existence and uniqueness of the solution to a system of nonlinear fully coupled forward-backward doubly stochastic differential equations with Poisson jumps. Our system is Markovian in the sense that initial and terminal values depend on solutions, and are not just fixed random variables. We establish under some monotonicity conditions, the existence and uniqueness of strong solutions of such equations by using a continuation method.

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“…These results have been extended to mean-field systems in [3][4][5]. See also [2,12,32] for versions of the relaxed stochastic maximum principle including doubly forward-backward stochastic differential equations and stochastic equations driven by G-Brownian motion.…”

Section: The Relaxed Stochastic Maximum Principle and Contributions O...mentioning

confidence: 99%

On the maximum principle for relaxed control problems of nonlinear stochastic systems

Mezerdi,

Mezerdi

2024

Adv Cont Discr Mod

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We consider optimal control problems for a system governed by a stochastic differential equation driven by a d-dimensional Brownian motion where both the drift and the diffusion coefficient are controlled. It is well known that without additional convexity conditions the strict control problem does not admit an optimal control. To overcome this difficulty, we consider the relaxed model, in which admissible controls are measure-valued processes and the relaxed state process is governed by a stochastic differential equation driven by a continuous orthogonal martingale measure. This relaxed model admits an optimal control that can be approximated by a sequence of strict controls by the so-called chattering lemma. We establish optimality necessary conditions, in terms of two adjoint processes, extending Peng’s maximum principle to relaxed control problems. We show that relaxing the drift and diffusion martingale parts directly as in deterministic control does not lead to a true relaxed model as the obtained controlled dynamics is not continuous in the control variable.

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Necessary and Sufficient Optimality Conditions for Relaxed and Strict Control of Forward-Backward Doubly SDEs with Jumps Under Full and Partial Information

Cited by 4 publications

References 36 publications

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

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

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

On the maximum principle for relaxed control problems of nonlinear stochastic systems

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