Description 
In the talk we consider a variant of the basic problem of the
calculus of variations, where the Lagrangian
is convex and subject to randomness adapted to a Brownian filtration. We
solve
the problem by reducing it, via a limiting argument, to an unconstrained
control problem
that consists in finding an absolutely continuous process minimizing the
expected sum
of the Lagrangian and the deviation of the terminal state from a given
target position.
Using the Pontryagin maximum principle one can characterize a solution
of the unconstrained
control problem in terms of a fully coupled forwardbackward stochastic
differential equation
(FBSDE). We use the method of decoupling fields for proving that the
FBSDE has
a unique solution.
The talk is based on joint work with Alexander Fromm, Thomas Kruse and
Alexandre Popier.
