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

Abstract

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

Existence of an optimal control for a system driven by a degenerate coupled forward–backward stochastic differential equations

“…When the coefficients are uniformly Lipschitz and σ is non-degenerate, the existence and the uniqueness of solutions were established in [5] for equation (1); in this case, the existence of an optimal control was recently established in [3] when the coefficients σ and b are independent of z, and σ is independent of the control u. The case where b depends on z and σ is independent of z and u, the existence of an optimal control can be performed as in [3]. In the case where σ depends upon (x, y, u), the problem of the existence of an optimal control seems difficult to obtain by the method we developed here.…”

Model reduction and uncertainty quantification of multiscale diffusions with parameter uncertainties using nonlinear expectations

“…The theory and the stochastic calculus for G-SDE have been developed by Peng and co-workers [39,12]. Relevant preliminary work on existence and uniqueness of fully coupled FBSDE, G-FBSDE and the corresponding dynamic programming (Hamilton-Jacobi Bellman or HJB) equations is due to Redjil & Choutri [42] and Kebiri et al [4,3,2,26], showing the existence of a relaxed control based on results of El-Karoui et al [25].…”

“…[18,Ch. 7]), that the slow process R = R converges pathwise to a limit process that is the solution of the (here: deterministic) initial value problem (3) dr dt = F (r; θ) , r(0) = r , where…”