Principe de maximum pour des équations di¤erentielles stochastiques progressives retrogrades

Abstract

In this work, we derive a stochastic maximum principle for optimal stochastic control of systems driven by forward-backward stochastic di¤erential equations (FBSDEs in short) without controlled di¤usion. Our Objectif is to minimize : J (u ( )) = E [ (y (0)] ; such that 8 >>>>>>>< >>>>>>>: dx(t) = f (t; x(t); u(t)) dt + (t; x(t))dW(t); dy(t) = g (t; x(t); y(t); z(t); u(t)) dt + z(t)dW(t) X (0) = x0; y(T) = h(x (T) In this work, the control domain is not assumed to be convex. The proof of the main result is based on spike perturbation and Itô formula. This resultat has been developped by Wensheng Xu (1995). Stochastic maximum principle for optimal control problem of forward and backward system. The Journal of the Australian Mathematical Society. Series B. Applied Mathematics, 37, pp 172-185. Keywords. Forward-backward stochastic di¤erential equations, optimal stochastic control, Stochastic maximum principle. perturbation forte.