Stochastic control under partial observation
We study and revisit the optimal control problem of partially observed stochastic systems. By using a control randomization method, we provide a backward stochastic differential equation (BSDE) representation for the value function in a general framework including path-dependence in the coefficients (both on the state and control) and without any non degeneracy condition on the diffusion coefficient. In the standard Markovian case, this BSDE representation has important implications: it allows us to obtain 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, and is derived by means of the recent notion of differentiability with respect to probability measures introduced by P.L. Lions in his lectures on mean-field games at the Collège de France. An important feature of our approach is that it does not require any condition to guarantee existence of a density for the filter process solution to the controlled Zakai equation, as usually done for the separated problem. We give an explicit solution to our HJB equation in the case of a partially observed non Gaussian linear quadratic model. Finally, if time permitting, we discuss the issue of numerical treatment of the proposed randomized BSDE for solving partial observation control problem.
Based on joint works with E. Bandini (Luiss University), A. Cosso (Politecnico Milano), and M. Fuhrman (Universita di Milano).