Optimal dividend payout problem under both diffusion risk and Poisson risk in finite horizon

In this paper, we study an optimal dividend payout problem of a firm facing with both diffusion risk and Poisson risk. Mathematically, we need to solve a parabolic variational inequality involving an integro-differential operator with gradient constraint. Assuming the Poisson intensity is less than a given constant $ \lambda_0 $, we prove the existence, uniqueness, and monotonicity of a classical solution to the variational inequality without putting any constraint on the loss distribution function. The properties such as existence and strict monotonicity and the upper bound of the free boundary are also obtained.