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

Summary We give a probabilistic interpretation of the Monte Carlo scheme proposed by Fahim, Touzi and Warin [Ann. Appl. Probab. 21(4) : 1322-1364 (2011)] for fully nonlinear parabolic PDEs, and hence generalize it to the path-dependent (or non-Markovian) case for a general stochastic control problem. General convergence result is obtained by weak convergence method in spirit of Kushner and Dupuis [19]. We also get a rate of convergence using the invariance principle technique as in Dolinsky [7], 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.