One of the primary methods for solving optimization problems involves determining a set of necessary conditions for any optimal solution. These conditions can become sufficient under additional convexity assumptions on the objective/constraint functions. Pontryagin's Maximum Principle (PMP) expands this idea to optimal control problems, i.e., optimization problems in infinite dimensional spaces. Namely, PMP states that for an optimally controlled dynamical system, there exists an adjoint set of equations; together, they both solve a two-point boundary value problem plus a maximum condition for a function called the Hamiltonian. In this talk, we present a stochastic PMP for dynamical systems affected by a measure-valued random environment, such that the state process is assumed to be driven by a diffusion with self-exciting jumps. Then, we apply our results to the problem of Mean-Variance Portfolio Selection with Regime Switching. 

This is a joint work with Daniel Hernández Hernández (CIMAT).