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Acta Mathematica Sinica,English Series 2021年第3期37卷 389-407页

作者： Jing Hai SHAO Kun ZHAO Center for Applied Mathematics Tianjin UniversityTianjin 300072P.R.China

This work is concerned with the continuous dependence on initial values of solutions of stochastic functional differential equations(SFDEs) with state-dependent regime-switching. Due to the state-dependence, this problem is very different to the corresponding problem for SFDEs without switching or SFDEs with Markovian switching. We provide a method to overcome the intensive interaction between the continuous component and the discrete component based on a subtle application of Skorokhod’s representation for jumping processes. Furthermore, we establish the strong convergence of Euler–Maruyama’s approximations, and estimate the order of error. The continuous dependence on initial values of Euler–Maruyama’s approximations is also investigated in the end.

关键词： Regime-switching state-dependent Euler–Maruyama’s approximation

Ergodicity and First Passage Probability of Regime-Switching Geometric Brownian Motions

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Chinese Annals of Mathematics,Series B 2018年第4期39卷 739-754页

作者： Jinghai SHAO Genter for Applied Mathematics Tianjin University Tianjin 300072 China.

A regime-switching geometric Brownian motion is used to model a geometric Brownian motion with its coefficients changing randomly according to a Markov *** this work, the author gives a complete characterization of the recurrent property of this process. The long time behavior of this process such as its p-th moment is also studied. Moreover, the quantitative properties of the regime-switching geometric Brownian motion with two-state switching are investigated to show the difference between geometric Brownian motion with switching and without switching. At last, some estimates of its first passage probability are established.

关键词： Ergodicity Regime-switching diffusions Lyapunov functions First passageprobability

Heavy tail and light tail of Cox-Ingersoll-Ross processes with regime-switching

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Science China Mathematics 2020年第6期63卷 1169-1180页

作者： Tongtong Hou Jinghai Shao School of Mathematical Sciences Beijing Normal UniversityBeijing 100875China Center for Applied Mathematics Tianjin UniversityTianjin 300072Chind

This study aims to investigate the tail behavior of Cox-Ingersoll-Ross(CIR) processes with regime switching. An essential difference shown in this study between CIR processes with and without regime switching is that the stationary distribution of those with regime switching may be heavy-tailed. We first provide sharp criteria to justify the existence of a stationary distribution for the CIR process with regime switching, which is applied to study the long-term returns of interest rates. Then, we provide a criterion to identify whether this distribution is heavy-tailed. Our results provide theoretical evidence of the existence of regime switching for interest-rate models based on empirical evidence of a heavy-tailed distribution.

关键词： regime switching heavy tail light tail ergodicity

Pointwise characterizations of curvature and second fundamental form on Riemannian manifolds

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Science China Mathematics 2018年第8期61卷 1407-1420页

作者： Fengyu Wang Bo Wu Center for Applied Mathematics Tianjin University Department of Mathematics Swansea University School of Mathematical Sciences Fudan University

Let M be a complete Riemannian manifold possibly with a boundary?*** any C^1-vector field Z,by using gradient/functional inequalities of the(reflecting)diffusion process generated by L:=?+Z,pointwise characterizations are presented for the Bakry-Emery curvature of L and the second fundamental form of?M if it *** characterizations extend and strengthen the recent results derived by Naber for the uniform norm‖RicZ‖∞on manifolds without boundaries.A key point of the present study is to apply the asymptotic formulas for these two tensors found by the first author,such that the proofs are significantly simplified.

关键词： curvature second fundamental form diffusion process path space

Derivative formulas and Poincaré inequality for Kohn-Laplacian type semigroups

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Science China Mathematics 2016年第2期59卷 261-280页

作者： WANG Feng-Yu School of Mathematical Sciences Beijing Normal University Department of Mathematics Swansea University

As a generalization to the heat semigroup on the Heisenberg group, the diffusion semigroup generated by the subelliptic operator L :=1/2 sum from i=1 to m X_i^2 on R^(m+d):= R^m× R^d is investigated, where X_i(x, y) = sum (σki?xk) from k=1 to m+sum (((A_lx)_i?_(yl)) from t=1 to d,(x, y) ∈ R^(m+d), 1 ≤ i ≤ m for σ an invertible m × m-matrix and {A_l}_1 ≤ l ≤d some m × m-matrices such that the Hrmander condition *** first establish Bismut-type and Driver-type derivative formulas with applications on gradient estimates and the coupling/Liouville properties, which are new even for the heat semigroup on the Heisenberg group; then extend some recent results derived for the heat semigroup on the Heisenberg group.

关键词： Kohn-Laplacian type operator derivative formula Poincare inequality reverse Poincare inequality

Logarithmic Sobolev,Isoperimetry and Transport Inequalities on Graphs

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Acta Mathematica Sinica,English Series 2016年第10期32卷 1221-1236页

作者： Yu Tao MA Ran WANG Li Ming WU School of Mathematical Sciences & Lab.Math.Com.Sys. Beijing Normal University School of Mathematics and Statistics Wuhan University School of Mathematical Sciences University of Science and Technology of China Institute of Applied Math. Chinese Academy of Sciences Laboratoire de Math.CNRS-UMR 6620 Université Blaise Pascal

In this paper, we study some functional inequalities （such as Poincar@ inequality, loga- rithmic Sobolev inequality, generalized Cheeger isoperimetric inequality, transportation-information inequality and transportation-entropy inequality） for reversible nearest-neighbor Markov processes on connected finite graphs by means of （random） path method. We provide estimates of the involved constants.

关键词： Graph log-Sobolev inequality isoperimetry transport inequality

First passage probabilities of one-dimensional diffusion processes

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Frontiers of Mathematics in China 2015年第4期10卷 901-916页

作者： Huijie JI Jinghai SHAO School of Mathematical Sciences Beijing Normal University Beijing 100875 China College of Mathematics and Computer Science Shanxi Normal UniversityLinfen 041000 China

This work is devoted to calculating the first passage probabilities of one-dimensional diffusion processes. For a one-dimensional diffusion process, we construct a sequence of Markov chains so that their absorption probabilities approximate the first passage probability of the given diffusion process. This method is especially useful when dealing with time-dependent boundaries.

关键词： Boundary crossing probability first passage probability Markovchain Skorokhod approximation

Transportation Cost Inequalities for Stochastic Reaction-Diffusion Equations with Lévy Noises and Non-Lipschitz Reaction Terms

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Acta Mathematica Sinica,English Series 2020年第2期36卷 121-136页

作者： Yu Tao MA Ran WANG School of Mathematical Sciences & Lab. Math. Com. Sys. Beijing Normal University School of Mathematics and Statistics Wuhan University

For stochastic reaction-diffusion equations with Levy noises and non-Lipschitz reaction terms,we prove that W\H transportation cost inequalities hold for their invariant probability measures and for their process-leve... 详细信息

关键词： Stochastic reaction-diffusion equation poisson random measure transportation cost in-equality

Variational formula for the stability of regime-switching diffusion processes

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Science China Mathematics 2018年第4期61卷 659-676页

作者： Jinghai Shao Lingdi Wang Center for Applied Mathematics Tianjin University School of Mathematics and Statistics Henan University

The asymptotical stability in probability is studied for diffusion processes and regime-switching diffusion processes in this work. For diffusion processes, some criteria based on the integrability of the functionals of the coefficients are given, which yield a useful comparison theorem on stability with respect to some nonlinear systems. For regime-switching diffusion processes, some criteria based on the idea of a variational formula are given. Both state-independent and state-dependent regime-switching diffusion processes are investigated in this work. These conditions are easily verified and are shown to be sharp by examples.

关键词： stability in probability regime-switching diffusions state-dependent M-matrix

Moderate Deviations for Stochastic Models of Two-Dimensional Second-Grade Fluids Driven by Lévy Noise

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Communications in Mathematics and Statistics 2018年第4期6卷 583-612页

作者： Wuting Zheng Jianliang Zhai Tusheng Zhang School of Mathematical Sciences University of Science and Technology of ChinaHefei 230026China School of Mathematics University of ManchesterOxford RoadManchester M139PLUK

In this paper,we establish a moderate deviation principle for stochastic models of two-dimensional second-grade fluids driven by Lévy *** will adopt the weak convergence *** of the appearance of jumps,this result is ... 详细信息

关键词： Moderate deviations Second-grade fluids Lévy process Weak convergence method