一个梯度约束的变分不等式中的自由边界问题

doi:10.3969/j.issn.1000-5641.2018.01.001

华东师范大学学报(自然科学版) ›› 2018, Vol. 2018 ›› Issue (1): 1-10.doi: 10.3969/j.issn.1000-5641.2018.01.001

• 数学 • 下一篇

一个梯度约束的变分不等式中的自由边界问题

管崇虎¹, 陈婧²

1. 嘉应学院数学学院, 广东梅州 514015,;
2. 香港中文大学系统工程与工程管理学院, 香港 999077

收稿日期:2017-02-15 出版日期:2018-01-25 发布日期:2018-01-11
通讯作者: 陈婧,女,博士研究生,研究方向为金融工程.E-mail:jchen@se.cuhk.edu.hk. E-mail:jchen@se.cuhk.edu.hk
作者简介:管崇虎,男,讲师,博士,研究方向为金融数学.E-mail:316346917@qq.com.
基金资助:
国家自然科学基金，（11626117，11471276，11601163）；广东省自然科学基金（2016A030307008）

A free boundary problem in variational inequality with gradient constraint

GUAN Chong-hu¹, CHEN Jing²

1. School of Mathematics, Jiaying University, Meizhou Guangdong 514015, China;
2. School of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, Hong Kong 999077, China

Received:2017-02-15 Online:2018-01-25 Published:2018-01-11

摘要/Abstract

摘要： 考虑一个抛物型梯度约束的变分不等式
min{vt-1/2σ²v_xx-μv_x+cv，v_x-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.

关键词: 梯度约束, 变分不等式, 自由边界, 随机最优控制, 最优分红

Abstract: Consider a parabolic variational inequality with gradient constraint
min{vt-1/2σ²v_xx-μv_x+cv,v_x-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.

Key words: gradient constraint, variational inequality, free boundary, stochastic optimal control, optimal dividend

中图分类号:

O157.26

管崇虎, 陈婧. 一个梯度约束的变分不等式中的自由边界问题[J]. 华东师范大学学报(自然科学版), 2018, 2018(1): 1-10.

GUAN Chong-hu, CHEN Jing. A free boundary problem in variational inequality with gradient constraint[J]. Journal of East China Normal University(Natural Sc, 2018, 2018(1): 1-10.

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一个梯度约束的变分不等式中的自由边界问题

A free boundary problem in variational inequality with gradient constraint

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