Idempotent Expansions for Continuous Time Stochastic Control
Max plus methods have previously been used to solve deterministic control problems. The methods are based on max plus (or min plus) expansions and can yield curse of dimensionality free numerical methods. In this paper, we explore min plus methods for continuous time stochastic control on a finite time horizon. We first approximate the original value function via time discretization. By generalizing the min plus distributive property to continuum spaces, we obtain an algorithm for recursive computation of the time discretized values, which we refer to as the idempotent distributed dynamic programming principle (IDDPP) Under the IDDPP, the value function at each step can be represented as an infimum of functions in a certain class. This is a min plus expansion for the value function. For the specific class of problems considered here, we see that the class can be taken as that consisting of the quadratic functions. A means for reducing the numbers of constituent quadratic functions is discussed.
SIAM J. Control. Optim.