Discussion of dynamic programming and linear programming approaches to stochastic control and optimal stopping in continuous time

This paper seeks to highlight two approaches to the solution of stochastic control and optimal stopping problems in continuous time. Each approach transforms the stochastic problem into a deterministic problem. Dynamic programming is a well-established technique that obtains a partial/ordinary differential equation, variational or quasi-variational inequality depending on the type of problem; the solution provides the value of the problem as a function of the initial position (the value function). The other method recasts the problems as linear programs over a space of feasible measures. Both approaches use Dynkin’s formula in essential but different ways. The aim of this paper is to present the main ideas underlying these approaches with only passing attention paid to the important and necessary technical details.