An Iterative Simulation Approach for Solving Stochastic Control Problems in Finance

Abstract

We develop a numerical method that combines functional approximations and dynamic programming to solve high-dimensional discrete-time stochastic control problems under general constraints. The method employs quasi-random grids and radial basis functions to handle high-dimensional state spaces; Lagrange multipliers to incorporate constraints and obtain first order optimality conditions; and a new algorithm that iteratively approximates conditional expectations of the value function with second order polynomials and shrinking approximation regions --- the test region iterative contraction algorithm (TRIC). We provide asymptotic results for the convergence and the speed of convergence of the TRIC algorithm, and apply the method to two Finance applications: a) dynamic portfolio choice with labor income and financial constraints, a continuous control problem; b) dynamic portfolio choice with capital gain taxation, a high-dimensional singular control problem.