Title: On Zhou's maximum principle for near-optimal control of mean-field forward-backward stochastic systems with jumps and its applications
Authors: Mokhtar Hafayed; Abdelmadjid Abba; Samira Boukaf
Addresses: Laboratory of Applied Mathematics, Biskra University, P.O. Box 145, Biskra 07000, Algeria ' Laboratory of Applied Mathematics, Biskra University, P.O. Box 145, Biskra 07000, Algeria ' Faculty of Economics Sciences, El Oued University, El Oued 039000, Algeria
Abstract: This paper is concerned with stochastic maximum principle for near-optimal control of nonlinear controlled mean-field forward-backward stochastic systems driven by Brownian motions and random Poisson martingale measure (FBSDEJs in short) where the coefficients depend on the state of the solution process as well as on its marginal law through its expected value. Necessary conditions of near-optimality are derived where the control domain is non-convex. Under some additional hypotheses, we prove that the near-maximum condition on the Hamiltonian function in integral form is a sufficient condition for "-optimality. Our result is derived by using the spike variation method, Ekeland's variational principle and some estimates of the state and adjoint processes, along with Clarke's generalised gradient for non-smooth data. This paper extends the results obtained by Zhou (1998) to a class of mean-field stochastic control problems involving mean-field FBSDEJs. As an application, mean-variance portfolio selection mixed with a recursive utility functional optimisation problem is discussed to illustrate our theoretical results.
Keywords: stochastic control; near-optimal control; mean-field forward-backward stochastic systems; differential equations; jumps; necessary conditions; sufficient conditions; near-optimality; variational principle; maximum principle; nonlinear control; Brownian motion; random Poisson martingale; portfolio selection; recursive utility functional optimisation.
International Journal of Modelling, Identification and Control, 2016 Vol.25 No.1, pp.1 - 16
Received: 27 Feb 2015
Accepted: 01 May 2015
Published online: 21 Jan 2016 *