Regression Methods for Stochastic Control Problems

In this paper we develop several regression algorithms for solving general stochastic optimal control problems via Monte Carlo. This type of algorithms is particularly useful for problems with high-dimensional state space and complex dependence structure of the underlying Markov process with respect to some control. The main idea of the algorithms is to simulate a set of trajectories under some reference measure and to use a dynamic program formulation combined with fast methods for approximating conditional expectations and functional optimizations on these trajectories. Theoretical properties of the presented algorithms are investigated and convergence to the optimal solution is proved under mild assumptions. Finally, we present numerical results showing the efficiency of regression algorithms in a case of a high-dimensional Bermudan basket options, in a model with a large investor and transaction costs.

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