Mesures Martingales et Application au Contrôle Stochastique

In this work, we are interested in the necessary conditions of optimality in stochastic optimal control whose system is controlled by a martingale measure. These necessary conditions will be established with theorems of approximations. In the first chapter, we were interested in certain definitions, properties and the construction of martingales measures and then we give results of extension of this problem. In the second chapter, one considers theorems of approximation of martingales measures by stochastic integrals with respect to the Brownian motion and convergence within the meaning of L2 between the initial and the relaxed problems. In the final chapter, one defines the problem of relaxed stochastic controls, and the relation between the weak solutions of the stochastic differential equations and the martingales problems. Then one defines the control rules and then we prove an existence result