First-Order Pontryagin Maximum Principle for Risk-Averse Stochastic Optimal Control Problems

Abstract

In this paper, we derive a set of first-order Pontryagin optimality conditions for a risk-averse stochastic optimal control problem subject to final time inequality constraints, and whose cost is a general finite coherent risk measure. Unlike previous contributions in the literature, our analysis holds for classical stochastic differential equations driven by standard Brownian motions. Moreover, it presents the advantages of neither involving second-order adjoint equations, nor leading to the so-called weak version of the PMP, in which the maximization condition with respect to the control variable is replaced by the stationarity of the Hamiltonian.

Viability and Exponentially Stable Trajectories for Differential Inclusions in Wasserstein Spaces

“…In [21,27], based on this definition of differential inclusions, we proved analogues in the setting of Wasserstein spaces of the Filippov estimates and Peano existence theorem, as well as a relaxation principle and a compactness criterion for the solution sets. These fundamental results are known to be extremely useful to investigate the fine properties of optimal control problems, both in the classical deterministic [58,80] and stochastic [16] settings, while enjoying natural generalisations to study e.g. evolution equations in Banach spaces [55,57] or mutational dynamics in metric spaces [13,56].…”