The Relaxed Stochastic Maximum Principle in Singular Optimal Control of Diffusions

Abstract

We consider a stochastic control problem where the set of controls is not necessarily convex and the system is governed by a nonlinear backward stochastic differential equation. We establish necessary as well as sufficient conditions of optimality for two models. The first concerns the strict (classical) controls. The second is an extension of the first to relaxed controls, who are a measure valued processes.where b is given function, ξ is the terminal data and W = (W t ) t≥0 is a standard d-dimensional Brownian motion, defined on a filtered probability space Ω, F , (F t ) t≥0 , P satisfying the usual conditions. The control variable v = (v t ), called strict (classical) control, is an F t -adapted process with values in some set U of R k . We denote by U the class of all strict controls.The criteria to be minimized, over the set U, has the formLemma 13 (Ekeland's variational principle). Let (E, d) be a complete metric space and f : E −→ R be lower-semicontinuous and bounded from below. Given ε > 0, suppose u ε ∈ E satisfies f (u ε ) ≤ inf (f ) + ε. Then for any λ > 0, there exists v ∈ E such that

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“…The second-order stochastic maximum principle for nonlinear SDEs with a controlled diffusion matrix was obtained by Bahlali and Mezerdi [19], extending the Peng maximum principle [20] to singular control problems. A similar approach has been used by Bahlali et al in [21], to study the stochastic maximum principle in relaxed-singular optimal control in the case of uncontrolled diffusion. Bahlali et al in [22] discuss the stochastic maximum principle in singular optimal control in the case where the coefficients are Lipschitz continuous in , provided that the classical derivatives are replaced by the generalized ones.…”