Pointwise second-order necessary conditions for stochastic optimal controls, Part II: The general case

TL;DR

This paper extends second-order necessary conditions for stochastic optimal controls to more general cases, including nonconvex control regions and controls affecting both drift and diffusion, using advanced variational and adjoint equations.

Contribution

It introduces a comprehensive framework with multiple variational and adjoint equations to establish second-order necessary conditions in the most general stochastic control setting.

Findings

Derived second-order necessary conditions for nonconvex control regions

Established Pontryagin-type maximum principle for stochastic singular controls

Extended the theory to controls influencing both drift and diffusion

Abstract

This paper is the second part of our series of work to establish pointwise second-order necessary conditions for stochastic optimal controls. In this part, we consider the general cases, i.e., the control region is allowed to be nonconvex, and the control variable enters into both the drift and the diffusion terms of the control systems. By introducing four variational equations and four adjoint equations, we obtain the desired necessary conditions for stochastic singular optimal controls in the sense of Pontryagin-type maximum principle.