A Free Boundary Problem Arising from a Stochastic Optimal Control Model with Bounded Dividend Rate.

We consider a Barenblatt parabolic equationArising from a financial stochastic optimal control model. In this model, the control variablel, which is bounded and lies in [0,M], should be chosen to optimize the objective function to take the maximum value. From the problem, it can be seen thatlshould be either 0 orM, which depends on whethervxis greater than 1 or not. We divide the domain into two parts, {vx> 1} and {vx⩽ 1}. Thus, the junction of the two regions, that is, free boundary, has particular financial implications. It can be expressed as a functional formh(t). In this article, we not only prove the existence and uniqueness of the solution to this equation, but we also study the property of the free boundaryh(t). We show thath(t) is a differentiable, nondecreasing function. We also present the shapes ofh(t) in different cases. The most difficult point is to prove the concavity of the value function by stochastic analysis. [ABSTRACT FROM AUTHOR]