| In this thesis, we consider the problem of portfolio optimization under jump-diffusion models in an incomplete market, where the counting processes in the jumps are correlated through a common shock component. Under the criterion of maximiz-ing the expected utility of terminal wealth, we adopt a martingale approach to solve this optimization problem. The equivalent martingale measure is the key of solving financial problems by martingale approach, because it can ensure the discounted asset prices become local martingales under the new measure. Considering that the equiv-alent martingale measure is not unique in an incomplete market, we find the optimal one by solving the dual problem in Theorem 3.2.2. and characterize the optimal port-folio in terms of a system of nonlinear equations regarding the optimal martingale measure in Theorem 3.2.1. Furthermore, we apply the results to the case of pow-er utility and obtain explicit solutions for optimal portfolios, and then we prove the existence and uniqueness of the optimum solution. However, we find the stochastic control approach is simpler in solving the optimal portfolio problems and has more applications than the martingale approach in incomplete markets. At the end of this article, we give some numerical examples to show the impact of model parameters on the decisions. |
