Decision and Control Laboratory
Coordinated Science Laboratory
Mean Field Approach to Stochastic Control with Partial Information
Dr. Alain Bensoussan
Lars Magnus Ericsson Chair
University of Texas at Dallas
Wednesday, October 2, 2019
3:00pm – 4:00pm
CSL Auditorium (B02)
The classical stochastic control problem under partial information can be formulated as a control problem for Zakai equation, whose solution is the unnormalized conditional probability distribution of the state of the system, which is not directly accessible. Zakai equation is a stochastic Fokker-Planck equation. Therefore, the mathematical problem to be solved is very similar to that met in Mean Field Control theory. Since Mean Field Control theory is much posterior to the development of Stochastic Control with partial information, the tools, techniques and concepts obtained in the last decade, for Mean Field Games and Mean field type Control theory, have not been used for the control of Zakai equation. It is the objective of this work to connect the two theories. Not only, we get the power of new tools, but also we get new insights for the problem of stochastic control with partial information. For mean field theory, we get new interesting applications, but also new problems. The possibility of using direct methods is, of course, quite fruitful. Indeed, if Mean Field Control Theory is a a very comprehensive and powerful framework , it leads to very complex equations, like the Master equation, which is a nonlinear infinite dimensional P.D.E., for which general theorems are hardly available, although an active research in this direction is performed. Direct methods are particularly useful to obtain regularity results. We will develop in detail the linear quadratic regulator problem, but because we cannot just consider the gaussian case, well know results , like the separation principle are not available. An interesting and important result is available in the literature, due to A. Makowsky. It describes the solution of Zakai equation for linear systems with general initial condition (non-gaussian). Curiouly, this result had not been exploited for the control aspect, in the literature. We show that the separation principle can be extended for quadratic pay-off functionals, but the Kalman filter is much more complex than in the gaussian case.
Since joining UT Dallas in 2004, Dr. Alain Bensoussan has been devoted to developing risk and decision analysis into a scientific field. He became interested in studying risk after his experiences in the space sector. The issue of risk is inherent in many areas, including engineering, finance, security or natural hazards. As such, Bensoussan is helping to build a new interdisciplinary science that can provide models and tools for a large variety of applications.
“I wanted to devote my time to developing risk management as a scientific field because it is important to have in mind that risks, wherever they come from, have common features,” he said. “Therefore, I approach risks by finding concepts, methods and techniques which can apply in a similar way, whether it’s in finance, nuclear energy or any other domain.”
He is head of UT Dallas’ International Center for Decision and Risk Analysis, which develops risk management research as it pertains to large-investment industrial projects that involve new technologies, applications and markets. The National Science Foundation, the state of Texas and European agencies support his research.
He is the co-editor of Risk and Decision Analysis and a fellow of the Institute of Electrical and Electronics Engineers and Society for Industrial and Applied Mathematics. He is also a Von Humboldt Research Award recipient.
Bensoussan served as president of the National Institute for Research in Computer Science and Control. He is chair professor of Risk and Decision Analysis at the Hong Kong Polytechnic University and World Class University Distinguished Professor at Ajou University. He is Professor Emeritus at the University of Paris Dauphine.
He earned a master’s degree in mathematics and sciences from Ecole Polytechnique, a master’s degree in economics and statistics from Ecole Nationale de la Statistique et de l’Administration Economique, and a PhD in mathematics from the University of Paris.
Bensoussan was selected by the Society of Industrial and Applied Mathematics to receive the 2014 W.T. and Ida Reid Prize for his contributions in the fields of differential equations and control theory. In 2013, he was named a Fellow of the American Mathematical Society.
Bensoussan is intensifying his collaboration with the alternative energy industry in the domain of wind power forecasting.