报 告 人：许顺吉教授

报告题目：On HJB Equations for Consumption Problems

报告时间：2015年4月26日（周日下午3:30）

报告地点：江苏师范大学数学与统计学院学术报告厅（静远楼1506室）

主办单位:数学与统计学院、科技处

报告摘要：

Merton-type  optimal consumption problems are classical portfolio optimization  problems. There is a huge literature addressing the problem, although  its study is still far from complete. We can view the problem as a  stochastic control problem. The dynamic programming is a standard  approach used in the control theory, which leads to the  Hamilton-Jacobi-Bellman equation (HJB). When the HJB equation can be  solved, a candidate of (Markovian) optimal consumption policy can be  derived. This approach suggests that an optimal consumption problem can  be solved through a study of a nonlinear partial differential equation  (elliptic type or parabolic equation), hence a need for a serious study  of HJB equation.

In  this note, we consider the general factor models as an example to show  some calculation based on HJB equation and martingale method. The first  part is a calculation for complete market. In the second part we  consider incomplete market. The use of “fictitious” completion in  martingale leads to a game interpretation of HJB equation. This suggests  a solution of HJB equation by choosing a direction to the saddle point  (of the game problem). This reduces the problem to the problems on a  sequence of complete markets. The convergence will be discussed. The  idea may have practical use to construct approximation of optimal  portfolio. The discussions relate to recent studies of Rogers (2003),  Rogers- Zaczkowski (2013). The talk is based on joint works with Hata  (2012),  Hata-Nagai-Sheu (2015) and an ongoing project of  Fleming-Nagai-Sheu.

许顺吉教授简介：

Shuenn-Jyi  Sheu received B.S. from Department of Mathematics, National Central  University in 1975 and M.S. from National Taiwan University in 1977. In  1979, he came to Brown University for the graduate study in Mathematics  and received his Ph.D. in 1982. Afterward he was a visiting assistant  professor in the Division of Applied Mathematics, Brown University for  one year. In 1983, he joined the Institute of Mathematics, Academia  Sinica as an Associate Research Fellow. From 1986, he is a Research  Fellow .

Dr.  Sheu major research interest is the applications of stochastic  analysis. In the past few years, he has done some works on the problems  related to the large deviation theory as well as on the use of control  method to study the problems related to Markov processes including  diffusion processes. He also works on some mathematical problems arised  from Monte Carlo method. He has nearly 50 publications  in Ann. Probab.，Ann. Appl. Probab.，Math. Finance，Probab. Theory Related Fields，SIAM J. Math. Anal.，SIAM J. Control Optim.，Appl. Math. Optim.，Trans.， Amer. Math. Soc..