Financial valuation and optimal strategy for retirement benefits in a jump diffusion model

We consider a defined benefit pension plan with the option of early retirement in a jump-diffusion model. The retirement benefits depend on the salary at the time of retirement, but with guaranteed minimum. The underlying salary follows a jump-diffusion process. In this note, we study the financial valuation of the retirement benefits and discuss the optimal retirement strategy. The valuation of retirement benefits can be characterized as the solution of an optimal stopping time problem. It is also related to a variational inequality or a free boundary problem of an integro-differential operator of the parabolic type. We prove that the valuation of the retirement benefits is the unique solution of the variational inequality and derive some properties of the free boundary, which correspond to the optimal retirement strategy.