Dynamic Investment Strategies to Reaction–Diffusion Systems Based upon Stochastic Differential Utilities

We study the dynamic investment strategies in continuous-time settings based upon stochastic differential utilities of Duffie and Epstein (Econometrica 60:353–394, 1992). We assume that the asset prices follow interacting Ito-Poisson processes, which are known to be the so-called reaction–diffusion systems. Stochastic maximum principle for stochastic control problems described by some backward-stochastic differential equations that are driven by Poisson jump processes allows us to derive the optimal investment strategies as well as optimal consumption. We shall furthermore propose a numerical procedure for solving the associated nested quasi-linear partial differential equations.