Randomized filtering and Bellman equation in Wasserstein space for partial observation control problem

We study a stochastic optimal control problem for a partially observed diffusion. By using the
control randomization method in Bandini et al. (2018), we prove a corresponding randomized dynamic
programming principle (DPP) for the value function, which is obtained from a flow property of an associated
filter process. This DPP is the key step towards our main result: a characterization of the value function
of the partial observation control problem as the unique viscosity solution to the corresponding dynamic
programming Hamilton–Jacobi–Bellman (HJB) equation. The latter is formulated as a new, fully non linear
partial differential equation on the Wasserstein space of probability measures. An important feature of
our approach is that it does not require any non-degeneracy condition on the diffusion coefficient, and no
condition is imposed to guarantee existence of a density for the filter process solution to the controlled
Zakai equation. Finally, we give an explicit solution to our HJB equation in the case of a partially observed
non Gaussian linear–quadratic model.