Numerical methods for optimal dividend payment and investment strategies of Markov-modulated jump diffusion models with regular and singular controls

This work focuses on numerical methods for finding optimal dividend payment and investment policies to maximize the present value of the cumulative dividend payment until ruin; the surplus is modeled by a regime-switching jump diffusion process subject to both regular and singular controls. Using the dynamic programming principle, the optimal value function obeys a coupled system of nonlinear integro-differential quasi-variational inequalities. Since the closed-form solutions are virtually impossible to obtain, we use Markov chain approximation techniques to approximate the value function and optimal controls. Convergence of the approximation algorithms are proved. Examples are presented to illustrate the applicability of the numerical methods.