### Backward Stochastic Differential Equations and Optimal Switching Problem with Two-Time-Scale Markov Chains: Weak Convergence and Stock Trading

This talk is concerned with backward stochastic differential equations (BSDEs) and optimal switching problem coupled by a continuous-time finite-state Markov chain which has a two-time-scale structure, i.e., the states of the Markov chain can be divided into a number of groups such that the chain jumps rapidly within a group and slowly between the groups. In this talk, we give some convergence results as the fast jump rate goes to infinity, which can be used to reduce the complexity of the original problem. This method is also referred to as singular perturbation.

The first result is the weak convergence of the BSDEs with two-time-scale Markov chains. It is shown that the solution of the original BSDE system converges weakly under the Meyer-Zheng topology to that of an aggregated BSDE system. Then we focus on the optimal switching problem for regime switching model with two-time-scale Markov chains. We obtain the optimal switching strategy by virtue of dynamic programming principle and prove the convergence of the value function under the two-time-scale structure. Finally, as an application of the theoretical results, an example concerning the stock trading problem in a regime switching market is provided. Both the optimal trading rule and convergence result are numerically demonstrated in this example.

吴臻，山东大学数学学院教授、博士生导师。国家杰出青年基金获得者，教育部“长江学者”特聘教授，国家“万人计划”首批科技创新领军人才入选者，享受国务院政府特殊津贴，科技部首批国家创新人才推进计划 “金融数学”重点领域创新团队负责人，研究领域涉及概率论、控制论和金融数学等，主要研究方向为正倒向随机微分方程与随机最优控制理论及其在金融中的应用。入选2002年度教育部优秀青年教师资助计划和2004年度教育部首批新世纪优秀人才支持计划，2004年获霍英东高校青年教师基金奖励资助，2005年被评为山东省优秀青年知识分子，2008年获第八届山东省青年科技奖，2008年获山东省自然科学首届杰出青年基金，2009年入选山东省有突出贡献的中青年专家，2015年入选国家百千万人才工程并获得“有突出贡献中青年专家”荣誉称号。现任国际控制理论权威期刊SIAM Journal on Control and Optimization和SCI学术期刊Statistics & Probability Letters编委，联合国教科文组织国际理论物理中心 (ICTP) 合作研究员 (Regular Associate)。获得国家教学成果二等奖两项和山东省教学成果一等奖两项，二等奖一项，被评为山东省第四届优秀研究生指导教师，主持负责一门国家精品课程。被聘为第七届国家自然科学基金数学天元基金学术领导小组成员，担任中国数学会理事, 中国概率统计学会常务理事,中国自动化学会控制理论专业委员会委员，山东数学会副理事长，被聘为山东省人民政府金融工作办公室咨询专家。