Nested variational inequalities and related optimal starting-stopping problems

This paper introduces several versions of starting-stopping problem for the diffusion model defined in terms of a stochastic differential equation. The problem could be regarded as a stochastic differential game in which the player can only decide when to start the game and when to quit the game in order to maximize one’s fortune. Nested variational inequalities arise in studying such a problem, with obtained the value function can be characterized and optimal strategies obtained.