A multilevel approach for stochastic nonlinear optimal control

We consider a class of finite-time horizon nonlinear stochastic optimal control problem. Although the optimal control admits a path integral representation for this class of control problems, efficient computation of the associated path integrals remains a challenging task. Wepropose a new Monte Carlo approach that significantly improves upon existing methodology. We tackle the issue of exponential growth in variance with the time horizon by casting optimal control estimation as a smoothing problem for a state-space model, and applying smoothing algorithms based on particle Markov chain Monte Carlo. To further reduce the cost, we then develop a multilevel Monte Carlo method which allows us to obtain an estimator of the optimal control with O(epsilon(2)) mean squared error with a cost of O(epsilon(-2) log(epsilon)(2)). In contrast, a cost of O(epsilon(-3)) is required for the existing methodology to achieve the same mean squared error. Our approach is illustrated on two numerical examples.