Discrete-time probabilistic approximation of path-dependent stochastic control problems

TL;DR

This paper extends a Monte Carlo scheme for fully nonlinear PDEs to handle path-dependent stochastic control problems, providing convergence proofs, rate estimates, and an implementable simulation-regression approach.

Contribution

It generalizes existing Monte Carlo methods to non-Markovian control problems with theoretical convergence and practical simulation techniques.

Findings

Established a probabilistic interpretation for the scheme.

Proved convergence and derived a rate of convergence.

Developed an implementable simulation-regression scheme.

Abstract

We give a probabilistic interpretation of the Monte Carlo scheme proposed by Fahim, Touzi and Warin [Ann. Appl. Probab. 21 (2011) 1322-1364] for fully nonlinear parabolic PDEs, and hence generalize it to the path-dependent (or non-Markovian) case for a general stochastic control problem. A general convergence result is obtained by a weak convergence method in the spirit of Kushner and Dupuis [Numerical Methods for Stochastic Control Problems in Continuous Time (1992) Springer]. We also get a rate of convergence using the invariance principle technique as in Dolinsky [Electron. J. Probab. 17 (2012) 1-5], which is better than that obtained by viscosity solution method. Finally, by approximating the conditional expectations arising in the numerical scheme with simulation-regression method, we obtain an implementable scheme.