Existence of optimal controls for systems governed by mean-field stochastic differential equations

Abstract

In this paper, we study the existence of an optimal control for systems, governed by stochastic dierential equations of mean-eld type. For non linear systems, we prove the existence of an optimal relaxed control, by using tightness techniques and Skorokhod selection theorem. The optimal control is a measure valued process dened on another probability space. In the case where the coecients are linear maps and the cost functions are convex, we prove by using weak convergence techniques, the existence of an optimal strict control, adapted to the initial ltration. Resume. Dans cet article on s'interesse a l'existence d'un controle optimal, pour des systemes gouvernes par des equations differentielles  stochastiques de type champ moyen. Pour les systemes non lineaires, on demontre un resultat d'existence d'un controle optimal relaxe, en utilisant des techniques de tension et le theoreme de selection de Skorokhod. Le controle optimal obtenu est un processus a valeurs mesures, defini sur un autre espace de probabilite. Dans le cas ou les coefficients sont lineaires et les fonctions de cout sont convexes, on demontre en utilisant des   techniques de convergence faible, l'existence d'un contrele optimal strict, adapte a la filtration initiale. Key words : Mean-field; Stochastic differential equation; Relaxed control; Existence; Tight-ness; Weak convergence.