Well Posedness of Operator Valued Backward Stochastic Riccati Equations in Infinite Dimensional Spaces

TL;DR

This paper establishes existence and uniqueness of solutions for operator-valued backward stochastic Riccati equations in infinite-dimensional spaces, with applications to stochastic control problems.

Contribution

It provides the first rigorous proof of well-posedness for these equations and applies the results to control theory in infinite dimensions.

Findings

Proved existence and uniqueness of solutions.

Characterized the value function and optimal feedback law.

Utilized semigroup regularization properties.

Abstract

We prove existence and uniqueness of the mild solution of an infinite dimensional, operator valued, backward stochastic Riccati equation. We exploit the regularizing properties of the semigroup generated by the unbounded operator involved in the equation. Then the results will be applied to characterize the value function and optimal feedback law for a infinite dimensional, linear quadratic control problem with stochastic coefficients.