On the backward stochastic Riccati equation in infinite dimensions

We study backward stochastic Riccati equations (BSREs) arising in quadratic optimal control problems with infinite dimensional stochastic differential state equations. We allow the coefficients, both in the state equation and in the cost, to be random. In such a context BSREs are backward stochastic differential equations living in a non-Hilbert space and involving quadratic nonlinearities. We propose two different notions of solutions to BSREs and prove, for both of them, existence and uniqueness results. We also show that such solutions allow to perform the synthesis of the optimal control. Finally we apply our results to the optimal control of a delay equation and of a wave equation with random damping.