Pontryagin Principle for Random Process

Question

Can Pontryagin's principle be written for minimizing the expected cost in the case of a stochastic process which is controlled? (cost at each time is function of state and action)?? I have not come across any such formulation.

Answer 1

Somewhat dual method to PMP is called the dynamic programming (DP), in particular see the following Bellman optimality principle. The rough idea is to split the total cost into the current part and the future part (which are often in the trade-of). It applies both to the deterministic systems, and stochastic systems. For the discrete time case see any book of Bertsekas, especially the one with Shreve, and for the continuous time case see any book on controlled diffusions, e.g. that one.