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

Abstract

We consider a Barenblatt parabolic equation Arising from a financial stochastic optimal control model. In this model, the control variable l, 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 that l should be either 0 or M, which depends on whether v_x is greater than 1 or not. We divide the domain into two parts, {v_x > 1} and {v_x ⩽ 1}. Thus, the junction of the two regions, that is, free boundary, has particular financial implications. It can be expressed as a functional form h(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 boundary h(t). We show that h(t) is a differentiable, nondecreasing function. We also present the shapes of h(t) in different cases. The most difficult point is to prove the concavity of the value function by stochastic analysis.

Keywords:

• Parabolic Barenblatt Equation
• Stochastic optimal control
• Controllable strategy
• Free boundary

Mathematics Subject Classification (2010):

• 35R35
• 60G40
• 91B70
• 93E20

Additional information

Funding

The project is supported by NNSF of China (No.11271143 and No.11371155) and University Special Research Fund for Ph.D. Program of China (20124407110001 and 20114407120008).