The uniqueness of the solution for the definite problem of a parabolic variational inequality

The uniqueness of the solution for the definite problem of a parabolic variational inequality is proved. The problem comes from the study of the optimal exercise strategies for the perpetual executive stock options with unrestricted exercise in financial market. Because the variational inequality is degenerate and the obstacle condition contains the partial derivative of an unknown function, it makes the theoretical study of the definite problem of the variational inequality problem very difficult. Firstly, the property which the value function satisfies is derived by applying the Jensen inequality. Then the uniqueness of the solution is proved by using this property and maximum principles.