  We consider a class of finite time horizon nonlinear stochastic optimal
control problem, where the control acts additively on the dynamics and the
control cost is quadratic. This framework is flexible and has found
applications in many domains. 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 Monte Carlo
task. The focus of this article is to propose a new Monte Carlo approach that
significantly improves upon existing methodology. Our proposed methodology
first tackles 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 associated with the control problem, and applying smoothing algorithms
based on particle Markov chain Monte Carlo. To further reduce computational
cost, we then develop a multilevel Monte Carlo method which allows us to obtain
an estimator of the optimal control with $\mathcal{O}(\epsilon^2)$ mean squared
error with a computational cost of
$\mathcal{O}(\epsilon^{-2}\log(\epsilon)^2)$. In contrast, a computational cost
of $\mathcal{O}(\epsilon^{-3})$ is required for existing methodology to achieve
the same mean squared error. Our approach is illustrated on two numerical
examples, which validate our theory.
