In this seminar we treat a class of semilinear parabolic differential equation in an infinite dimensional Hilbert space, that in particular includes Hamilton Jacobi Bellman (HJB in the following) equations arising in stochastic optimal control problems. Namley we consider HJB equations related to problems satisfying the so called “structure conditons”, meaning that the control affects the system only through the noise. We show that wave equations and delay equations satisfy this “structure condition”.
We introduce the notion of mild solution for HJB equations, and we show how to prove existence and uniquess of a mild solution for HJB equations both by fixed point arguments and by a pure probabilistic approach based on Backward Stochastic Differential Equations.