*Journées de Probabilités 2007*

**Giuseppina Guatteri **

*Backward stochastic Riccati equations in Hilbert spaces *

In this talk we consider stochastic control problems for linear systems with random coefficients taking values in a Hilbert space *H*. We treat quadratic cost functionals. We follow the dynamic programming approach and we introduce the Riccati equations that in this case is a backward stochastic equation in the space of linear and bounded operators from *H* to *H* with nonlinearity of quadratic type. We cope with the fact that the space of linear and bounded operators is no more an Hilbert space as soon as *H* is infinite dimensional. We consider both the finite and the infinite horizon case. Our result can be applied to a wide class of stochastic PDEs.