勷勤数学•专家报告

题      目：Stochastic control problems with state-reflections arising from relaxed benchmark tracking

报  告  人：薄立军  教授  (邀请人：杨舟)

                                  西安电子科技大学

时      间：11月9日  14：30-15：30

地     点：数科院西楼二楼会议室

报告人简介:

       薄立军教授本科毕业于西安电子科技大学数学系，硕士和博士毕业于南开大学概率论与数理统计专业，研究方向为随机分析与金融数学。2012年入选教育部新世纪优秀人才支持计划、陕西国家应用数学中心交叉团队负责人、陕西省杰青；主持国家自然科学基金数理学部青年项目1项、面上项目3项、陕西数理基础科学研究重点项目、中科院前沿科学重点研究计划-青年拔尖科学家项目；获陕西省工业与应用数学学会首届青年科技奖(2019年)和2023年度陕西高等学校科学技术研究优秀成果奖特等奖(第一完成人); 目前已在国际公认的金融数学、管理和运筹学以及保险精算领域权威期刊Math. Finan., Finan. & Stoch., SIAM J. Finan. Math., SIAM J. Control & Optim, Math. Opers. Res., J. Banking & Finan., JEDC，IME和Quant. Finan.上发表学术论文70余篇。

摘      要：

       This talk discusses stochastic control problems motivated by optimal consumption with wealth benchmark tracking. The benchmark process is modeled by a combination of a geometric Brownian motion and a running maximum process, indicating its increasing trend in the long run. We consider a relaxed tracking formulation such that the wealth compensated by the injected capital always dominates the benchmark process. The stochastic control problem is to maximize the expected utility of consumption deducted by the cost of the capital injection under the dynamic floor constraint. By introducing two auxiliary state processes with reflections, an equivalent auxiliary control problem  is formulated and studied, which leads to the HJB equation with two Neumann boundary conditions. We establish the existence of a unique classical solution to the dual PDE using some novel probabilistic representations involving the local time of some dual processes together with a tailor-made decomposition-homogenization technique. The proof of the verification theorem on the optimal feedback control can be carried out by some stochastic flow analysis and technical estimations of the optimal control.

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