  We describe an adaptive importance sampling algorithm for rare events that is
based on a dual stochastic control formulation of a path sampling problem.
Specifically, we focus on path functionals that have the form of cumulate
generating functions, which appear relevant in the context of, e.g.~molecular
dynamics, and we discuss the construction of an optimal (i.e. minimum variance)
change of measure by solving a stochastic control problem. We show that the
associated semi-linear dynamic programming equations admit an equivalent
formulation as a system of uncoupled forward-backward stochastic differential
equations that can be solved efficiently by a least squares Monte Carlo
algorithm. We illustrate the approach with a suitable numerical example and
discuss the extension of the algorithm to high-dimensional systems.
