Dynamic Utility and related nonlinear SPDE driven by Lévy Noise.

Summary This work concerns the study of consistent dynamic utilities in a financial market with jumps. We extend the results established in the paper [EKM13] to this framework. The ideas are similar but the difficulties are different due to the presence of the Lévy process. An additional complexity is clearly the interpretation of the terms of jumps in the different problems primal and dual one and relate them to each other. To do, we need an extension of the Itô-Ventzel's formula to jump's frame. By verification, we show that the dynamic utility is solution of a non-linear second order stochastic partial integro-differential equation (SPIDE). The main difficulty is that this SPIDE is forward in time, so there are no results in the literature that ensure the existence of a solution or simply allow us to deduce important properties, in our study, such as concavity or monotonicity. Our approach is based on a complete study of the primal and the dual problems. This allows us, firstly, to establish a connection between the utility-SPIDE and two SDEs satisfied by the optimal processes. Based on this connection and the SDE's theory, stochastic flow technics and characteristic method allow us, secondly, to completely solve the equation.