Integral operator Riccati equations arising in stochastic Volterra control problems

TL;DR

This paper proves existence and uniqueness of infinite-dimensional Riccati equations in the context of stochastic Volterra control problems, extending classical matrix Riccati theory to a broader functional setting.

Contribution

It introduces a novel framework for solving Riccati equations in Banach spaces involving signed matrix measures, applicable to stochastic Volterra equations.

Findings

Established existence and uniqueness of solutions

Extended Riccati theory to infinite-dimensional Banach spaces

Applicable to stochastic Volterra control problems

Abstract

We establish existence and uniqueness for infinite dimensional Riccati equations taking values in the Banach space L 1 ($\mu$ $\otimes$ $\mu$) for certain signed matrix measures $\mu$ which are not necessarily finite. Such equations can be seen as the infinite dimensional analogue of matrix Riccati equations and they appear in the Linear-Quadratic control theory of stochastic Volterra equations.