Stochastic Control with Perfect Measurements

where L(t) is a given matrix. Unlike the deterministic setting, when the state is a random variable one cannot directly use the variational optimality conditions since the adjoint equation λ̇ = −∂xH cannot describe all possible random evolutions of x(t). We must resort to dynamic programming. We next consider dynamic programming for stochastic systems through the derivation of the stochastic Hamilton-Jacobi-Bellman equation. The value function V (x, t) computed over the time interval [t, tf ] is defined by