Design of a Pure Endowment Life Insurance Contract Based on Optimal Stochastic Control

There are a variety of products in the life insurance literature. These products differ in how the benefits are paid and the execution time. In this paper, we designed pure-endowment insurance contract and obtain the optimal strategy and consumption for a policyholder with CRRA utility function. In our designed contract, premiums are received from the policyholder at certain times. The insurer undertakes to pay the premiums by a certain guarantee rate, in addition, by investing in a portfolio of risky and risk free assets and share invest profits. The optimal stochastic control method can be used in a financial market with a risk free asset and a risky stock asset with jump by infinite activity L\'{e}vy model. We employed Variance Gamma process as a representative of infinite activity jump models and sensitivity of jump parameters in an uncertainty financialmarket has been studied. Also we compared results using by two forces of mortality.