Stochastic Volterra equations with time-changed L\'evy noise and maximum principles

TL;DR

This paper develops maximum principles for control problems involving non-Markovian Volterra dynamics driven by time-changed Lévy noise, using backward stochastic differential equations and stochastic derivatives.

Contribution

It introduces a novel maximum principle framework for non-Markovian Volterra systems with time-changed Lévy noise, expanding control theory tools.

Findings

Established necessary and sufficient stochastic maximum principles.

Applied backward stochastic differential equations with time-change.

Extended stochastic control methods to non-Markovian processes.

Abstract

Motivated by a problem of optimal harvesting of natural resources, we study a control problem for Volterra type dynamics driven by time-changed L\'evy noises, which are in general not Markovian. To exploit the nature of the noise, we make use of different kind of information flows within a maximum principle approach. For this we work with backward stochastic differential equations (BSDE) with time-change and exploit the non-anticipating stochastic derivative introduced in [15]. We prove both a sufficient and necessary stochastic maximum principle.