一个梯度约束的变分不等式中的自由边界问题

摘要 考虑一个抛物型梯度约束的变分不等式 min{vt-1/2σ2vxx-μvx+cv，vx-1}=0. 问题来源于以公司最优分红模型为背景的随机最优控制问题.本文运用偏微分方程技术，通过引入惩罚方法，得到变分不等式解的存在唯一性和一些先验估计，然后进一步讨论自由边界的性质，最终证明自由边界不仅可以表示成x关于t的函数，而且是以0为起点、单调递增C∞的曲线.
Abstract：Consider a parabolic variational inequality with gradient constraint min{vt-1/2σ2vxx-μvx+cv,vx-1}=0. The problem stems from a stochastic optimal control problem based on optimal dividend model. By using PDE technique and the penalty method, the existence and uniqueness and some a priori estimates of the solution of the variational inequality are obtained, and then the properties of the free boundary are further discussed. It is proved that the free boundary can be expressed as a function of x with respect to t and is a monotonically increasing, C∞ smooth curve starting from zero.