design of a pure endowment life insurance contract based on optimal stochastic control

In this paper, we design a pure-endowment insurance contract and obtain the optimal strategy and consumption for a policyholder with crra utility function. in this contract, premiums are received from the policyholder at certain times. theinsurer undertakes to pay the premiums by a certain guarantee rate, in addition, by investing in a portfolio of risky and risk free assets share invest pro ts. we used variance gamma process as a representative of in nite activity jump modelsand sensitivity of jump parameters in an uncertainty  nancial market has been studied. also we compared results using by two forces of mortality.