On the Backward Stochastic Riccati Equation in Infinite Dimensions

This paper deals with linear quadratic stochastic control problems in infinite-dimensional Hilbert space with random operator valued coefficients. This is certainly a classical problem, first treated by J.-M. Bismut [SIAM J. Control Optimization 14 (1976), no. 3, 419–444; MR0406663 (53 #10449)] for general finite-dimensional problems and later by N. U. Ahmed [SIAM J. Control Optim. 19 (1981), no. 3, 401–430; MR0613102 (82j:93054)] for infinite-dimensional problems involving unbounded random operator valued coefficients. Following a basic variational principle one can easily derive the necessary conditions for optimality. This approach gives rise to backward stochastic differential equations on infinite-dimensional spaces. The questions of existence, even uniqueness, and regularity properties of solutions of the associated operator Riccati equations are of fundamental importance. Such problems have been treated in [N. U. Ahmed, op. cit.] for operators based on the Gel′fand triple {V, H, V ∗}. In the paper under review the operators include a deterministic infinitesimal generator of a C0 semigroup and some bounded linear operator valued stochastic processes. The paper presents some interesting results on the existence and uniqueness of solutions of the stochastic Riccati equations. Conceptually one may consider this as the solution of the LQR problem in infinite-dimensional Hilbert space, though from a practical point of view this is far from it. It seems the authors have missed some earlier papers in the same area which we included here (see the papers cited above as well as [N. U. Ahmed, in Differential equations and applications, Vol. I, II (Columbus, OH, 1988), 13–19, Ohio Univ. Press, Athens, OH, 1989; MR1026110 (91h:60067)]).