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Prof. Wanyang Dai
Nanjing University, China
A Unified FB-SPDE with Jumps vs. A System of Coupled FB-SDEs with Skew Reflections: Theory and Applications
We study four systems and their interactions. First, we formulate a unified system of coupled forward and backward stochastic partial differential equations (FB-SPDEs) with Levy jumps, which is vector-valued and whose drift, diffusion, and jump coefficients may involve partial differential operators. Under generalized local linear growth and Lipschitz conditions, the well-posedness concerning adapted strong solution to the FB-SPDEs is proved. Second, we consider a unified system of FB-SDEs, a special form of the FB-SPDEs, however, with skew reflections. Under generalized linear growth and Lipschitz conditions together with a general completely-S condition on reflection matrices, we prove the well-posedness of adapted weak solution to the FB-SDEs. In particular, if the spectral radii in certain sense for both reflection matrices are strictly less than the unity, a unique adapted strong solution will be concerned. Both the FB-SPDEs and the FB-SDEs cover their degenerated deterministic forms of FB-PDEs and FB-ODEs as special cases. Third, we formulate a stochastic differential game (SDG) problem with general number of players based on the FB-SDEs. By a solution to the FB-SPDEs, we determine a solution to the FB-SDEs under a given control rule and then obtain a Pareto optimal Nash equilibrium point to the non-zero-sum SDG problem. Fourth, we study the application of the FB-SPDEs in a queueing system and discuss how to use the queueing system to motivate the SDG problem.

Wanyang Dai received his Ph.D degree in applied mathematics jointly with industrial engineering and systems engineering from Georgia Institute of Technology, Atlanta, GA, U.S.A., in 1996, where he worked on stochastics and applied probability concerning network performance modeling and analysis, algorithm design and implementation via stochastic diffusion approximation. The breakthrough results and methodologies developed in his thesis were cited, used, and claimed as "contemporaneous and independent" achievements by some other subsequent breakthrough papers that were presented as "45 minute invited talk in probability and statistics" in International Congress of Mathematicians (ICM) 1998, which is the most privilege honor in the mathematical society. The finite element algorithm designed and implemented in the thesis to compute the stationary distributions of reflecting Brownian motions living in general-dimensional state space is also well-known to the related fields. 

He was a MTS (permanent) in End-to-End Network Architecture Department of AT&T Bell Labs and then (Alcatel-)Lucent Bell Labs/Network Systems, Atlanta, GA and Warren, NJ, U.S.A., from 1996-1999, where he was principal investigators and developers of several projects in telecommunication network architecture and design, network performance and financial engineering, operating system and database development, etc. with some project won "Technology Transfer". 

He is a full professor since 2004 in Nanjing University, where his research includes stochastic processes related optimization and optimal control, admission/scheduling/routing protocols and performance analysis/optimization for cloud computing and the next generation of wireless and wireline communication systems, forward/backward stochastic (ordinary/partial) differential equations and their applications to queueing systems, stochastic differential games, communication networks, financial engineering, etc. His achievements are published in "big name" journals including Operations Research and Queueing Systems. 

He is an Editor-in-Chief of International Journal of Information Engineering, editors or members of editorial boards of over 30 international journals ranging from pure mathematics to its applications, General Chairs, invited plenary/keynote speakers, and TPC members of over 20 IEEE and other international conferences, Vice Presidents and Standing Directors of several probability, statistics, and operations research societies. 

He was a visiting scholar in Academy of Mathematics and Systems Science of China, Beijing, China, in 2001, a Long Term Participant of IMA Annual Program of Probability and Statistics in Complex Systems: Financial Engineering, Communication Networks and Genomics in Institute of Mathematics and Its Applications (IMA), Minneapolis, U.S.A., in 2004, a visiting professor in Department of Systems Engineering and Engineering Management in Chinese University of Hong Kong, Hong Kong, China, in 2007.

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