Portfolio optimization in presence of a self-exciting jump process: from theory to practice

We aim at generalizing the celebrated portfolio optimization problem "à la Merton", where the asset evolution is steered by a self-exciting jump-diffusion process. We first define the rigorous mathematical framework needed to introduce the stochastic optimal control problem we are interesting in. Then, we provide a proof for a specific version of the Dynamic Programming Principle (DPP) with respect to the general class of self-exciting processes under study. After, we state the Hamilton-Jacobi-Bellman (HJB) equation, whose solution gives the value function for the corresponding optimal control problem. The resulting HJB equation takes the form of a Partial-Integro Differential Equation (PIDE), for which we prove both existence and uniqueness for the solution in the viscosity sense. We further derive a suitable numerical scheme to solve the HJB equation corresponding to the portfolio optimizationproblem. To this end, we also provide a detailed study of solution dependence on the parameters of the problem. The analysis is performed by calibrating the model on ENI asset levels during the COVID-19 worldwide breakout. In particular, the calibration routine is based on a sophisticated Sequential Monte Carlo algorithm.