  In this paper, we consider continuous-time stochastic optimal control
problems where the cost is evaluated through a coherent risk measure. We
provide an explicit gradient descent-ascent algorithm which applies to problems
subject to non-linear stochastic differential equations. More specifically, we
leverage duality properties of coherent risk measures to relax the problem via
a smooth min-max reformulation which induces artificial strong concavity in the
max subproblem. We then formulate necessary conditions of optimality for this
relaxed problem which we leverage to prove convergence of the gradient
descent-ascent algorithm to candidate solutions of the original problem.
Finally, we showcase the efficiency of our algorithm through numerical
simulations involving trajectory tracking problems and highlight the benefit of
favoring risk measures over classical expectation.
