Monte Carlo algorithms for optimal stopping and statistical learning

We extend the Longstaff.Schwartz algorithm for approximately solving optimal stopping problems on high-dimensional state spaces. We reformulate the optimal stopping problem for Markov processes in discrete time as a generalized statistical learning problem. Within this setup we apply deviation inequalities for suprema of empirical processes to derive consistency criteria, and to estimate the convergence rate and sample complexity. Our results strengthen and extend earlier results obtained by Clément, Lamberton and Protter [Finance and Stochastics 6 (2002) 449.471].