The purpose of this dissertation is to solve an optimal investment, consumption and life insurance problem described by jump-diffusion processes in two settings.
First, we consider a problem with random parameters of a wage earner who wants to save to his beneficiary for his death. Using one risk-free asset and one risky asset price given by a geometric jump-diffusion process, we obtain the optimal strategy via the dynamic programming approach, combining the Hamilton-Jacobi-Bellman equation with a backward stochastic differential equation with jumps.
Secondly, we discuss the optimal investment, consumption and life insurance problem with capital constraints. The problem consists of one risk-free asset and two risky asset prices defined in an independent Brownian motion and Poisson process. We derive the optimal strategy of the unconstrained problem via martingale approach, from which, the problem with capital constraint is solved applying the option based portfolio insurance method.