Mean Field approach to stochastic control with partial information

Alain Bensoussan, Sheung Chi Phillip Yam

Published 2019-09-23Version 1

The classical stochastic control problem under partial information can be formulated as a control problem for Zakai equation, whose solution is the unnormalized conditional probability distribution of the state of the system, which is not directly accessible. Zakai equation is a stochastic Fokker-Planck equation. Therefore, the problem to be solved is similar to that met in Mean Field Control theory. Since Mean Field Control theory is much posterior to the development of Stochastic Control with partial information, the tools, techniques, and concepts obtained in the last decade, for Mean Field Games and Mean field type Control theory, have not been used for the control of Zakai equation. It is the objective of this work to connect the two theories. Not only, we get the power of new tools, but also we get new insights for the problem of stochastic control with partial information. For mean field theory, we get new interesting applications, but also new problems. Indeed, Mean Field Control Theory leads to very complex equations, like the Master equation, which is a nonlinear infinite dimensional P.D.E., for which general theorems are hardly available, although active research in this direction is performed. Direct methods are useful to obtain regularity results. We will develop in detail the linear quadratic regulator problem, but because we cannot just consider the Gaussian case, well-known results, as the separation principle is not available. An interesting and important result is available in the literature, due to A. Makowsky. It describes the solution of Zakai equation for linear systems with general initial condition (non-gaussian). Curiously, this result had not been exploited for the control aspect, in the literature. We show that the separation principle can be extended for quadratic pay-off functionals, but the Kalman filter is much more complex than in the gaussian case.