First and second order necessary conditions for stochastic optimal controls

TL;DR

This paper derives first and second order necessary conditions for stochastic optimal controls with nonconvex control sets, using variational analysis and a minimal number of adjoint equations.

Contribution

It introduces a novel approach to establish necessary optimality conditions for stochastic controls with nonconvex control sets, using only one or two adjoint equations.

Findings

First order necessary conditions derived with one adjoint equation.

Second order necessary conditions established with two adjoint equations.

Applicable to control systems with control-dependent drift and diffusion terms.

Abstract

The main purpose of this paper is to establish the first and second order necessary optimality conditions for stochastic optimal controls using the classical variational analysis approach. The control system is governed by a stochastic differential equation, in which both drift and diffusion terms may contain the control variable and the set of controls is allowed to be nonconvex. Only one adjoint equation is introduced to derive the first order necessary condition; while only two adjoint equations are needed to state the second order necessary conditions for stochastic optimal controls.