A gradient descent-ascent method for continuous-time risk-averse optimal control

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 that 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 optimality conditions 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 expectations.