Existence of optimal controls for systems driven by FBSDEs

Bahlali, Khaled; Gherbal, Boulakhras; Mezerdi, Brahim

doi:10.1016/j.sysconle.2011.02.011

Cited by 29 publications

(35 citation statements)

References 15 publications

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“…By passing to the limit, we establish the existence of a relaxed optimal control to the initial problem.The existence of a strict control follows from the Filippov convexity condition. Our result improve in some sense those of [4,6].problem. This allows them to construct a sequence of optimal feedback controls.…”

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confidence: 80%

Existence of an optimal control for a coupled FBSDE with a non degenerate diffusion coefficient

et al. 2018

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We a controlled system driven by a coupled forward-backward stochastic differential equation (FBSDE) with a non degenerate diffusion matrix. The cost functional is defined by the solution of the controlled backward stochastic differential equation (BSDE), at the initial time. Our goal is to find an optimal control which minimizes the cost functional. The method consists to construct a sequence of approximating controlled systems for which we show the existence of a sequence of feedback optimal controls. By passing to the limit, we establish the existence of a relaxed optimal control to the initial problem.The existence of a strict control follows from the Filippov convexity condition. Our result improve in some sense those of [4,6].problem. This allows them to construct a sequence of optimal feedback controls. Then, they pass to the limit and use the result of [10] in order to get the existence of a relaxed optimal control. In both papers [4] and [6] the Filippov convexity condition is used in order to get the existence of a strict optimal control. It should be noted that in [4] and [6] the controlled system is driven by a decoupled FBSDE.The aim of the present paper is to extend the results of [4,6], to a coupled FBSDE. To begin, let us give a description of our problem.Let T > 0 be a finite horizon, t ∈ [0, T ] and (Ω, F, P, (F t )) be a filtered probability space satisfying the usual conditions. Let W be a d-dimensional Brownian motion with respect to the filtration (F t ). Let U be a compact metric space. We define the deterministicWe consider the following controlled coupled FBDSE defined for s ∈ [t, T ] by:dX t,x,u s = b(X t,x,u s , Y t,x,u s , u s )ds + σ(X t,x,u s , Y t,x,u s )dW s , X t,x,u t = x, dY t,x,u s = −f (X t,x,u s , Y t,x,u s , Z t,x,u s , u s )ds + Z t,x,u s dW s + dM t,x,u s , M t,x,u tSince V δ ′ is a classical solution to the Hamilton-Jacobi-Bellman equation it follows thatHence, the comparison Theorem shows thatUsing a symmetric argument, we deduce that :Since V δ and V δ ′ are deterministic, we haveHence, it suffices to estimate E(|Y δ ′ ,t ′ ,x ′ ,u δ t ′ − Y δ,t,x,u δ t |).We assume that t ′ < t, and for

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supporting

confidence: 80%

Existence of an optimal control for a coupled FBSDE with a non degenerate diffusion coefficient

et al. 2018

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“…Among other results, they prove existence of weak solutions with different methods and hypothesis (in particular the generator is assumed to be uniformly continuous in the space of variables) which ensure that the approximating sequence Z (n) constructed in their paper converges in L 2 to Z. Let us also mention the recent paper [4] on the existence of an optimal control for a FBSDE. This optimal control and the corresponding solutions are obtained by taking weak limits of minimizing controls and the corresponding strong solutions.…”

Section: Introductionmentioning

confidence: 99%

“…Remark 2.2 Similarly, a strong solution to (1) should be a triplet (Y, Z, L) defined on Ω × [0, T ]) satisfying (3), (4), and (1), and such that L is a càdlàg martingale orthogonal to W and L 0 = 0, but this notion coincides with that of a strong solution to (2), because then L would be an (F t )-martingale, hence L = 0.…”

Section: Introductionmentioning

confidence: 99%

Weak solutions of backward stochastic differential equations with continuous generator

Bouchemella¹,

Fitte²

2014

Stochastic Processes and their Applications

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This paper provides a simple approach for the consideration of quadratic BSDEs with bounded terminal We prove the existence of a weak solution to a backward stochastic differential equation (BSDE)in a finite-dimensional space, where f (t, x, y, z) is affine with respect to z, and satisfies a sublinear growth condition and a continuity condition. This solution takes the form of a triplet (Y, Z, L) of processes defined on an extended probability space and satisfyingwhere L is a martingale with possible jumps which is orthogonal to W . The solution is constructed on an extended probability space, using Young measures on the space of trajectories. One component of this space is the Skorokhod space D endowed with the topology S of Jakubowski.

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“…The study of stochastic optimal control initiated in the late of 1960s such as in [17] for finance. In recent decades, the characterization of the stochastic optimal control problem has been developed by many authors in literature, for example, see [1], [2], [4], [5], [8], [10], [22] and references therein. In addition to randomness, fuzziness is another important uncertainty, which plays an essential role in the real world.…”

Section: Introductionmentioning

confidence: 99%