Stochastic Optimal Control Based on Monte Carlo Simulation and Least-Squares Regressiond

Abstract

In the financial engineering field, many problems can be formulated as stochastic control problems. A unique feature of the stochastic control problem is that uncertain factors are involved in the evolution of the controlled system and thus the objective function in the stochastic control is typically formed by an expectation operator. There are in general two approaches to solve this kind of problems. One can reformulate the problem to be a deterministic problem and solve the corresponding partial differential equation. Alternatively, one calculates conditional expectations occurring in the problem by either numerical integration orMonte Carlo methods.
We focus on solving various types ofmulti-period stochastic control problems via the Monte Carlo approach. We employ the Bellman dynamic programming principle so that a multi-period control problem can be transformed into a composition of several singleperiod control problems, that can be solved recursively. For each single-period control problem, conditional expectations with different filtrations need to be calculated. In order to avoid nested simulation (i.e. Monte Carlo simulation within aMonte Carlo simulation), which may be very time consuming, we implement Monte Carlo simulation and cross-path least-squares regression. So-called “regress-later” and “bundling” approaches are introduced in our algorithms to make them highly accurate and robust. In most cases, high quality results can be obtained within seconds.