Portfolio optimization in a jump-diffusion market with durability and local substitution: A penalty approximation of a singular control problem

Abstract

The main purpose of this thesis is to study a singular finite-horizon portfolio optimization problem, and to construct a penalty approximation and numerical scheme for the corresponding Hamilton- Jacobi-Bellman (HJB) equation. The driving stochastic process of the portfolio optimization problem is a L evy process, and the HJB equation of the problem is a non-linear second order degenerate integro-partial dierential equation subject to gradient and state constraints. We characterize the value function of the optimization problem as the unique constrained viscosity solution of the HJB equation.

Our penalty approximation is obtained by studying a non-singular version of the original optimization problem. The original HJB equation is difficult to solve numerically because of the gradient constraint, and in the penalty approximation the gradient constraint is incorporated in the HJB equation as a penalty term. We prove that the solution of the penalty approximation converges to the solution of the original problem. We also construct a numerical scheme for the penalty approximation, prove convergence of the numerical scheme, and present results from numerical simulations.

Other new results of the thesis include some explicit solution formulas for our finite-horizon optimization problem, new constraints on the explicit time-independent solution formulas found in [8], and a lemma concerning the characterization of viscosity solutions by use of so-called subdifferentials and superdifferentials.