Peng's Maximum Principle for Stochastic Delay Differential Equations of Mean-Field Type

TL;DR

This paper extends Peng's maximum principle to stochastic delay differential equations of mean-field type, addressing control problems with delays, mean-field interactions, and non-convex control domains using an infinite-dimensional lifting approach.

Contribution

It introduces a novel maximum principle for mean-field stochastic delay differential equations with delays and non-convex controls, utilizing an infinite-dimensional Hilbert space framework.

Findings

Established a maximum principle for the extended class of equations.

Derived asymptotic estimates for the variational process.

Handled non-convex control domains in the stochastic delay setting.

Abstract

We extend Peng's maximum principle to the case of stochastic delay differential equations of mean-field type. More precisely, the coefficients of our control problem depend on the state, on the past trajectory and on its expected value. Moreover, the control enters the noise coefficient and the control domain may be non-convex. Our approach is based on a lifting of the state equation to an infinite dimensional Hilbert space that removes the explicit delay in the state equation. The main ingredient in the proof of the maximum principle is a precise asymptotic for the expectation of the first order variational process, which allows us to neglect the corresponding second order terms in the expansion of the cost functional.