Solving stochastic dynamic programs by convex optimization and simulation

In this chapter we review some recent progress on Monte Carlo methods for a class of stochastic dynamic programming equations, which accommodates optimal stopping problems and time discretization schemes for backward stochastic differential equations with convex generators. We first provide a primal maximization problem and a dual minimization problem, based on which confidence intervals for the value of the dynamic program can be constructed by Monte Carlo simulation. For the computation of the lower confidence bounds we apply martingale basis functions within a least-squares Monte Carlo implementation. For the upper confidence bounds we suggest a multilevel simulation within a nested Monte Carlo approach and, alternatively, a generic sieve optimization approach with a variance penalty term.