Forward Backward SDEs Systems for Utility Maximization in Jump Diffusion Models

We consider the classical problem of maximizing the expected utility of terminal net wealth with a final random liability in a simple jump-diffusion model. In the spirit of Horst et al. (Stoch Process Appl 124(5):1813–1848, 2014) and Santacroce and Trivellato (SIAM J Control Optim 52(6):3517–3537, 2014), under suitable conditions the optimal strategy is expressed in implicit form in terms of a forward backward system of equations. Some explicit results are presented for the pure jump model and for exponential utilities.