Integral Operator Riccati Equations Arising in 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.