Control problem on space of random variables and master equation

In this article, we study a control problem in an appropriate space of random variables; in fact, in our set up, we can consider an arbitrary Hilbert space, yet we specialize only to a Hilbert space of square-integrable random variables. We see that the control problem can then be related to a mean field type control problem. We explore here a suggestion of Lions in (Lectures at College de France, http://www.college-de-france.fr) and (Seminar at College de France). Mean field type control problems are control problems in which functionals depend on probability measures of the underlying controlled process. Gangbo and Święch [J. Differ. Equ. 259 (2015) 6573–6643] considered this type of problem in the space of probability measures equipped with the Wasserstein metric and use the concept of Wasserstein gradient; their work provides a completely rigorous treatment, but it is quite intricate, because metric spaces are not vector spaces. The approach suggested by Lions overcomes this difficulty. Nevertheless, our present proposed approach also benefits from the useful concept of L-derivatives as introduced in a recent interesting treatise of Carmona and Delarue [Probabilistic Theory of Mean Field Games with Applications. Springer Verlag (2017)]. We also consider Bellman equation and the Master equation of mean field type control. We provide also some extension of the results of Gangbo and Święch [J. Differ. Equ. 259 (2015) 6573–6643].