Existence of optimal controls for stochastic Volterra equations

TL;DR

This paper establishes conditions ensuring the existence of relaxed and strict optimal controls for stochastic Volterra equations, including cases with singular kernels, advancing the theoretical understanding of control problems involving complex stochastic systems.

Contribution

It provides new sufficient conditions for the existence of relaxed and strict optimal controls in stochastic Volterra equations with singular kernels, extending previous results.

Findings

Existence of relaxed optimal controls under specific integrability and growth conditions.

Conditions under which classical convexity assumptions lead to strict optimal controls.

Applicability to rough processes with singular kernels at zero.

Abstract

We provide sufficient conditions that guarantee the existence of relaxed optimal controls in the weak formulation of stochastic control problems for stochastic Volterra equations (SVEs). Our study can be applied to rough processes that arise when the kernel appearing in the controlled SVE is singular at zero. The existence of relaxed optimal policies relies on the interaction between integrability hypotheses on the kernel and growth conditions on the running cost functional and the coefficients of the controlled SVEs. Under classical convexity assumptions, we can also deduce the existence of optimal strict controls.