Sensitivity results in stochastic optimal control: A Lagrangian perspective^∗

¹ Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria.
julio.backhoff@univie.ac.at
² Institut de recherche XLIM-DMI, UMR-CNRS 7252 Faculté des sciences et techniques Université de Limoges, 87060 Limoges, France.
francisco.silva@unilim.fr

Received: 4 September 2014
Accepted: 16 July 2015

Abstract

In this work we provide a first order sensitivity analysis of some parameterized stochastic optimal control problems. The parameters and their perturbations can be given by random processes and affect the state dynamics. We begin by proving a one-to-one correspondence between the adjoint states appearing in a weak form of the stochastic Pontryagin principle and the Lagrange multipliers associated to the state equation when the stochastic optimal control problem is seen as an abstract optimization problem on a suitable Hilbert space. In a first place, we use this result and classical arguments in convex analysis, to study the differentiability of the value function for convex problems submitted to linear perturbations of the dynamics. Then, for the linear quadratic and the mean variance problems, our analysis provides the stability of the optimizers and the C¹-differentiability of the value function, as well as explicit expressions for the derivatives, even when the data perturbation is not convex in the sense of [R.T. Rockafellar, Conjugate duality and optimization. Society for Industrial and Applied Mathematics, Philadelphia, Pa. (1974).