Mean Field approach to stochastic control with partial information

Abstract

The classical stochastic control problem under partial information , as , for instance , described in the book of A. Bensoussan[2] , 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 mathematical problem to be solved is very 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 conncet 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.The possibility of using direct methods is , of course, quite fruitful.

Price Formation and Optimal Trading in Intraday Electricity Markets with a Major Player

“…Nevertheless, in Féron et al (2020), it has been shown that explicit solutions may be found in the mean field limit, where the number of agents is sent to infinity, and the influence of every single agent on the entire market becomes negligible. The use of mean field theory for stochastic control with partial information has also recently been proposed by Bensoussan and Yam (2019), in a formal fashion, in order to solve the associated Zakai equation.…”