Importance sampling in stochastic optimization: An application to intertemporal portfolio choice

Abstract

In this paper, we propose an approach to construct an analytical approximation of the zero-variance importance sampling distribution. We show specifically how this can be designed for the classic intertemporal portfolio choice problem with proportional transaction costs and constant relative risk aversion preferences. We compare the method to standard variance reduction techniques in single-period optimization and multi-stage stochastic programming formulations of the problem. The numerical experiments show that the method produces significant improvements in solution quality. In the single-period setting, the number of scenarios can be reduced by a factor of 400 with maintained solution quality compared to the best standard method; Latin hypercube sampling. Using importance sampling in multi-stage formulations, the gaps between lower and upper bound estimates are reduced by a factor of 26-500 with maintained scenario tree size. On a higher level, we consider analytical approximations of the zero-variance importance sampling distribution to be a promising method to improve solution quality in stochastic optimization.