Optimal investment and retirement strategies for defined-contribution pension with jump diffusion risk process

An optimal investment strategy and optimal delay retirement time of defined-contribution pension which want to achieve the expected utility maximization problem of the terminal wealth were studied in this paper. Firstly, the stochastic differential equation with jump-diffusion is constructed where the terminal time is free, to describe the corresponding asset accumulation process under unknown retirement time. Then, a stochastic optimal control problem with jump-diffusion risk process is proposed, where the delay retirement time and the investment strategy are control parameter and control function, respectively. In order to obtain the optimal solutions of above optimization model with Cobb-Douglas utility function, the stochastic optimal control problem with free terminal time is transformed to that with determined terminal time by time-scaling transformation. After that, based on stochastic control theory, a Hamilton-Jacobi-Bellman equation for stochastic optimal control problem with terminal time was derived. The optimal investment strategy is obtained by hypothesis method. Finally, sensitivity analysis and numerical simulation are performed.