### Rennes, 10 novembre 2014, amphi J Bât 28.

Programme

• 10h00 Accueil des participants en salle commune du bâtiment 22.

• 10h30 Marco Fuhrman (Politecnico di Milano) Optimal control of pure jump Markov processes and constrained backward stochastic differential equations
Résumé :  We consider an optimal stochastic control problem for general pure jump Markov processes described by their rate transition measure and we derive a probabilistic representation for the value function which is a solution to the corresponding Hamilton-Jacobi-Bellman (HJB) equation of integral type. The method we use, introduced in recent papers by I. Kharroubi, H Pham, J. Ma, J. Zhang, R. Elie, is based on a randomization of the control process. For the randomized system we introduce a "dual" control problem whose value function can be represented in terms of a constrained backward stochastic differential equations with partially nonpositive jumps. Finally we identify both value functions via analysis of the HJB equation. This is a joint work with Elena Bandini (Politecnico di Milano).

• 11h30 Cyril Labbé (University of Warwik) The Parabolic Anderson Model on R^3
Résumé : The theory of regularity structures allows one to define the solution to several ill-posed SPDEs, among which the Parabolic Anderson Model. So far, all these equations have been considered on a bounded spatial domain. In this work, we construct the solution to the Parabolic Anderson Model on the whole space R^3. I will explain how one needs to adapt the theory to achieve this construction. This is a joint work with Martin Hairer

12h45 : Déjeuner à proximité du campus

• 14h00 Loïc Chaumont (Université d'Angers) Transformation de Lamperti des processus de branchement multitypes
Résumé :  A partir de l'algorithme de recherche en largeur, nous  construisons une bijection entre l'ensemble des forêts planes multitypes et un certain ensemble de suites multivariées. Ce codage permet d'obtenir une transformation de type Lamperti entre les processus de branchement multitypes en temps discret et les marches aléatoires multivariées. Ces résultats sont ensuite utilisés pour l'étude en temps continu. Nous montrons en particulier que tout processus de branchement à d types, en temps continu et à valeurs discrètes peut être obtenu comme un processus Poisson d² dimensionnel changé de temps par l'intégrale de ce processus de branchement.

• 15h00 Marie-Amélie Morlais (Université du Maine) Existence and uniqueness of viscosity solutions for second order integro partial-differential equations without the monotonicity condition
Résumé : In this talk, we establish a new uniqueness result of a (continuous) viscosity solution for some integro partial-differential equation (or ipde in short). The novelty is that we relax the so-called monotonicity assumption on the driver, assumption which is classically assumed in the literature of viscosity solution for equation with a non local term. First , we explain the framework by providing the explicit form of the non linear ipde as well as the related BSDE with jumps. Then and in a second part, we proceed with the proof of our main result (existence and uniqueness of the viscosity solution in an appropriate class). For convenience, the proof is splitted into two main parts: the first one consists in establishing  precise estimates of the solution of the related BSDE with jumps. The second one uses this estimate to construct the unique viscosity solution of a standard ipde (with its non local term which is frozen). We then prove that this viscosity solution provides the solution of the ipde we are interested in (this relies both on a fixed point result and the Feynman Kac representation. Finally some straightforward generalizations (especially to the reflected case) are presented.