摘要：We are concerned with the linear-quadratic optimal stochastic control problem where  all the coefficients of  the control  system and   the running weighting matrices in the cost functional are allowed to be predictable (but essentially bounded) processes and the terminal state weighting matrix in the cost functional is allowed to be random.The fundamental problem was studied by J. M. Bismut [in SIAM J. Control & Optim., 14(1976), pp. 419--444 and in Seminaire de Probabilites XII, Lecture Notes in Math. 649, C. Dellacherie, P. A. Meyer, and M. Weil, eds., Springer-Verlag, Berlin, 1978, pp. 180--264] where he derived  the backward stochastic Riccati equation as the equivalent matrix form of the associated Bellman equation, while he left  its solution remain to be open in the general random case for a long time.  Subsequently  Shige Peng listed it [in Control of Distributed Parameter and Stochastic Systems (Hangzhou, 1998), S. Chen, et al., eds., Kluwer Academic Publishers, Boston, 1999, pp. 265-273] as an open problem for backward stochastic differential equations. The open problem was solved in 2003  [in \textit{SIAM J. Control \& Optim.}, 42(2003), pp.\ 53--75] via the stochastic maximum principle with a viewpoint of stochastic flow for the associated stochastic Hamiltonian system, and recently  [in SIAM J. Control & Optim., 53(2015), pp. 1082--1106] via the dynamic programming principle. The present talk will briefly introduce  the major ideas, mathematical tools  and techniques given in the last two papers. 

 
    个人简介：汤善健，复旦大学数学科学学院教授、博士生导师，主要从事随机控制和倒向随机方程的研究。２００３年用随机最大值原理、２０１５年用动态规划原理证明了由布朗运动驱动的一般倒向随机Riccati微分方程的解的存在唯一性，解决了法国科学院院士J. M. Bismut于197６年在SIAM J. Control Optimization 提出的问题；2000年在具极大秩的假定下，解决了Brockett在198３年国际数学家大会上的45分钟报告中提出的非线性滤波理论中的"有限维估计代数”的分类问题。1999年度德国“洪堡”学者，2003年度国家杰出青年基金获得者，2005年度长江特聘教授，2014年获得高等学校优秀科学研究成果自然科学一等奖。