Pointwise second-order necessary conditions for stochastic optimal controls, Part I: The case of convex control constraint

TL;DR

This paper develops pointwise second-order necessary conditions for stochastic optimal controls with convex control constraints, considering both drift and diffusion terms involving the control variable, under Malliavin calculus assumptions.

Contribution

It introduces the first set of pointwise second-order necessary conditions for stochastic controls with convex constraints, extending previous first-order results.

Findings

Established second-order necessary conditions for stochastic singular controls

Applicable to controls affecting both drift and diffusion terms

Relies on Malliavin calculus assumptions

Abstract

This paper is the first part of our series work to establish pointwise second-order necessary conditions for stochastic optimal controls. In this part, both drift and diffusion terms may contain the control variable but the control region is assumed to be convex. Under some assumptions in terms of Malliavin calculus, we establish the desired necessary condition for stochastic singular optimal controls in the classical sense.