A biographical note and tribute to xunjing li on his 80th birthday
Jin Ma Shige Peng Jiongmin Yong Xunyu Zhou
Mathematical Control & Related Fields 2015, 5(3): i-iii doi: 10.3934/mcrf.2015.5.3i
Professor Xunjing Li was born in Qingdao, Shandong Province, China, on the 13th June 1935. Shandong is a province with a rich culture that has nurtured a great number of influential intellectuals during its long history, including Confucius. Immediately after his graduation from the Department of Mathematics at Shandong University in 1956, Professor Li was enrolled into the master program at Fudan University specializing in function approximation theory, supervised by Professor Jiangong Chen, one of the most prominent Chinese mathematicians in modern history. He stayed at Fudan as an Assistant Lecturer upon graduation in 1959, and was promoted to Lecturer, Associate Professor and Professor in 1962, 1980, and 1984, respectively. He became a Chair Professor in 1997, before retiring in 2001. He died of cancer in February 2003 at the age of 68.

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Dynamic equilibrium limit order book model and optimal execution problem
Jin Ma Xinyang Wang Jianfeng Zhang
Mathematical Control & Related Fields 2015, 5(3): 557-583 doi: 10.3934/mcrf.2015.5.557
In this paper we propose a dynamic model of Limit Order Book (LOB). The main feature of our model is that the shape of the LOB is determined endogenously by an expected utility function via a competitive equilibrium argument. Assuming zero resilience, the resulting equilibrium density of the LOB is random, nonlinear, and time inhomogeneous. Consequently, the liquidity cost can be defined dynamically in a natural way.
    We next study an optimal execution problem in our model. We verify that the value function satisfies the Dynamic Programming Principle, and is a viscosity solution to the corresponding Hamilton-Jacobi-Bellman equation which is in the form of an integro-partial-differential quasi-variational inequality. We also prove the existence and analyze the structure of the optimal strategy via a verification theorem argument, assuming that the PDE has a classical solution.
keywords: optimal execution dynamic programming principle Limit order book verification theorem. viscosity solution liquidity risk
Pathwise Taylor expansions for Itô random fields
Rainer Buckdahn Ingo Bulla Jin Ma
Mathematical Control & Related Fields 2011, 1(4): 437-468 doi: 10.3934/mcrf.2011.1.437
In this paper we study the pathwise stochastic Taylor expansion, in the sense of our previous work [3], for a class of Itô-type random fields in which the diffusion part is allowed to contain both the random field itself and its spatial derivatives. Random fields of such an "self-exciting" type particularly contains the fully nonlinear stochastic PDEs of curvature driven diffusion, as well as certain stochastic Hamilton-Jacobi-Bellman equations. We introduce the new notion of "$n$-fold" derivatives of a random field, as a fundamental device to cope with the special self-exciting nature. Unlike our previous work [3], our new expansion can be defined around any random time-space point (τ,ξ), where the temporal component τ does not even have to be a stopping time. Moreover, the exceptional null set is independent of the choice of the random point (τ,ξ). As an application, we show how this new form of pathwise Taylor expansion could lead to a different treatment of the stochastic characteristics for a class of fully nonlinear SPDEs whose diffusion term involves both the solution and its gradient, and hence lead to a definition of the stochastic viscosity solution for such SPDEs, which is new in the literature, and potentially of essential importance in stochastic control theory.
keywords: stochastic characteristics. stochastic viscosity solutions Wick-square Pathwise stochastic Taylor expansion

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