Some results on pointwise second-order necessary conditions for stochastic optimal controls

TL;DR

This paper develops pointwise second-order necessary conditions for stochastic optimal controls, addressing cases where controls influence both drift and diffusion, with distinctions for convex and nonconvex control regions.

Contribution

It introduces novel second-order necessary conditions for stochastic controls, including singular controls, considering both convex and nonconvex control regions, and highlights the role of the second-order adjoint equation.

Findings

Second-order conditions depend on the control entering the diffusion term.

Correction terms from the second-order adjoint equation are essential.

Results apply to both convex and nonconvex control regions.

Abstract

The purpose of this paper is to derive some pointwise second-order necessary conditions for stochastic optimal controls in the general case that the control variable enters into both the drift and the diffusion terms. When the control region is convex, a pointwise second-order necessary condition for stochastic singular optimal controls in the classical sense is established; while when the control region is allowed to be nonconvex, we obtain a pointwise second-order necessary condition for stochastic singular optimal controls in the sense of Pontryagin-type maximum principle. It is found that, quite different from the first-order necessary conditions, the correction part of the solution to the second-order adjoint equation appears in the pointwise second-order necessary conditions whenever the diffusion term depends on the control variable, even if the control region is convex.