Stochastic optimal portfolios and life insurance problems in a Lévy market

Abstract

This thesis solves various optimal investment, consumption and life insurance problems described by jump-diffusion processes. In the first part of the thesis, we solve an optimal investment, consumption, and life insurance problem when the investor is restricted to capital guarantee. We consider an incomplete market described by a jump-diffusion model with stochastic volatility. Using the martingale approach, we prove the existence of the optimal strategy and the optimal martingale measure and we obtain the explicit solutions for the power utility functions. Secondly, we prove the sufficient and necessary maximum principle for the similar problem proposed in the first part. Then we apply the results to solve an investment, consumption, and life insurance problem with stochastic volatility, that is, we consider a wage earner investing in one risk-free asset and one risky asset described by a jump-diffusion process and has to decide concerning consumption and life insurance purchase. We assume that the life insurance for the wage earner is bought from a market composed of M > 0 life insurance companies offering pairwise distinct life insurance contracts. The goal is to maximize the expected utilities derived from the consumption, the legacy in the case of a premature death and the investor's terminal wealth. The third part discusses an optimal investment, consumption and insurance problem of a wage earner under inflation. Assume a wage earner investing in a real money account and three asset prices, namely: a real zero coupon bond, the inflation-linked real money account and a risky share described by jump-diffusion processes. Using the theory of quadratic-exponential backward stochastic differential equation (BSDE) with jumps approach, we derive the optimal strategy for the two typical utilities (exponential and power) and the value function is characterized as a solution of BSDE with jumps. The explicit solutions for the optimal investment in both cases of exponential and power utility functions for a diffusion case are derived.