Abstract. We study the relaxed optimal stochastic control problem for systems governed by stochastic differential equations (SDEs), driven by an orthogonal continuous martingale measure, where the control is allowed to enter both the drift and diffusion coefficient. The set of admissible controls is a set of measure-valued processes. Necessary conditions for optimality for these systems in the form of a maximum principle are established by means of spike variation techniques. Our result extends Peng's maximum principle to the class of measure valued controls.Résumé. Nousétudions les problèmes de contrôle stochastique relaxés pour des systèmes gouvernés par deséquations différentielles stochastiques (EDSs), dirigées par des mesures martingales orthogonales continues, avec un drift et un coefficient de diffusion contrôlé. L'ensemble des contrôles admissibles est constitué de processusà valeurs mesures. Onétablit des conditions nécessaires d'optimalité en utilisant des preturbations fortes. Notre résultat généralise le principe du maximum de Peng pour la classe de contrôlesà valeurs mesures.

We provide necessary optimality conditions for singular controlled stochastic differential equations driven by an orthogonal continuous martingale measure. The control is allowed to enter both the drift and diffusion coefficient and has two components, the first being relaxed and the second singular, the domain of the first control does not need to be convex, and for the relaxing method, we show by a counter-example that replacing the drift and diffusion coefficients by their relaxed counterparts does not define a true relaxed control problem. The maximum principle for these systems is established by means of spike variation techniques on the relaxed part of the control and a convex perturbation on the singular one. Our result is a generalization of Peng's maximum principle to singular control problems.

This paper deals with optimal combined singular and regular controls for stochastic Volterra integral equations, where the solution X u,ξ (t) = X(t) is given by