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### Open Access Journals

MCRF

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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MCRF

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.

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.

MCRF

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.
DCDS-S

Tourism load imbalance is a serious problem in Jiuzhaigou, which may destroy the environment and reduce the feeling of tourists. A reasonable initial shunt strategy is the key factor in making the scenic become balance. So, this paper considers this problem by finding an optimal initial shunt strategy. An optimization model containing two flow models is built, which takes the total overload balance as objective. In order to solve this model and find the optimal initial shunt strategy, a hGA is proposed as solution method. Finally, a comparison analysis is conducted to show the efficiency of this model in remitting the local overload problem.

CPAA

In this paper we describe and analyze the general
framework of cascading multigrid preconditioning. In particular, we
introduce two preconditioners based on the cascading multigrid
approach. We then illustrate the application of the corresponding
method to the heat equation with mild regularity in Besov spaces.
Furthermore, we analyze a cascading multigrid preconditioner for the
interior penalty discontinuous Galerkin method.

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