ON THE BACKWARD STOCHASTIC RICCATI EQUATION IN INFINITE DIMENSIONS Giuseppina Guatteri Dipartimento di Matematica, Politecnico di Milano piazza Leonardo da Vinci 32, 20133 Milano, Italia e-mail: giuseppina.guatteri@mate.polimi.it Gianmario Tessitore Dipartimento di Matematica, Universit` a di Parma, via D’Azeglio 85, 43100 Parma, Italia e-mail: gianmario.tessitore@unipr.it Abstract. 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 non-linearities. 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. 1. Introduction Backward stochastic Riccati differential equations (BSREs) naturally arise in the study of stochastic optimal linear quadratic control problems with stochastic coefficients. The interest of proving existence and uniqueness results for such a class of equations was firstly addressed by Bismut in [2]. It was clear from the beginning that to study those highly non-linear backward stochastic differential equations was a challenging task, already in the finite dimensional case (see [3], [21] or the historical review in [12]). The difficulty comes essentially from the fact that, in its general formulation, the BSRE involves quadratic terms in both the unknowns (in particular in the, so called, ‘martingale’ term). Moreover the non linearity can be well defined only in a subset of the space of non-negative matrices (where the equation naturally lives). Several works followed the pioneering paper [2] (see [20] [12], [13], [14] [15]). In particular only very recently, in [22], the proof of the existence and uniqueness of a solution of the BSRE was given in the general case corresponding to a finite dimensional, linear quadratic problem with random coefficients and state and control-dependent noise. This last result, somehow, completes the theory of finite dimensional BSREs. We remark that in all the above literature it is clear that the treatment of the equation can not be solely based on general backward stochastic differential equation techniques but needs to exploit the interplay between the Riccati equation and its control theoretic interpretation (for results on general backward stochastic differential equation with quadratic nonlinearities see [11] and [17]). On the other side several works, motivated by control of stochastic partial differential equations, have been devoted to linear quadratic optimal control problems for infinite dimensional stochastic differential equations with deterministic coefficients (see for instance [23] and references within). The corresponding Riccati equation is a deterministic nonlinear ODE in a suitable space of symmetric, non-negative, Hilbert valued operators. The present paper is, as far as we know, the first attempt to consider infinite dimensional BSREs. Such equations naturally arise in several models; namely they appear in all the situations in which one has to perform the synthesis of the optimal control for a linear quadratic problem having, as state equation, an infinite dimensional stochastic evolution equation with random coefficients (see examples in Sections 9 and 10). We also underline that the study of infinite dimensional BSREs introduces specific new difficulties in the theory of backward stochastic differential equations. Namely these are non-linear backward stochastic differential equations that involve unbounded linear terms and quadratic nonlinearities. Moreover, and this is the main difficulty, they naturally live in a non-Hilbertian infinite dimensional space. In order to separate difficulties we consider here only the case in which the non-linearity does not depend on the ’martingale term’ of the backward equation. In other words we consider the infinite dimensional analogue of the 1