Jicheng Liu

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Verified

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• PhD
• Professor (Full) at Huazhong University of Science and Technology

In this paper, we investigate the well-posedness of stochastic variational inequalities on [Formula: see text] with discontinuous drift coefficients. First, a case that drift coefficients are only integrable on [Formula: see text] is considered. Utilizing Girsanov’s theorem, Zvonkin’s transform and Yamada–Watanabe’s principle, we derive the weak we...

This paper investigates the nonparametric linear wavelet-based estimators of multivariate regression functions. Under mild conditions, we establish the asymptotic normality under the weak dependence, which incorporates mixing and association concepts. This framework applies to numerous classes of intriguing statistical processes, primarily Gaussian...

We develop the large deviations principle for synchronized system with small noise. Depending on the interaction between the intensity of the noise with coupling strength, we get different behavior. By simple transformations, the original synchronized system is equivalently converted into the slow-fast system, then we derive the representations for...

We prove a strong convergence rate of the averaging principle for general two-time-scales stochastic evolution equations driven by cylindrical Wiener processes. In particular, our general result can be used to deal with a large class of quasi-linear stochastic partial differential equations, such as stochastic reaction–diffusion equations, stochast...

The present article focuses on the nonparametric estimation of multivariate density and regression functions. We consider the nonparametric linear wavelet-based estimators and investigate the strong consistency from the theoretical viewpoint. In particular, we prove the strong uniform consistency properties of these estimators, over compact subsets...

In this paper, we mainly construct a connection between synchronized systems and multi-scale equations, and then use the averaging principle as an intermediate step to obtain synchronization. This strategy solves the synchronization problem of dissipative stochastic differential equations, regardless of the structure of the noise. Moreover, the ave...

This paper is devoted to the synchronization of stochastic lattice dynamical systems driven by fractional Brownian motion with Hurst parameter 12<H<1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemarg...

In the paper, an averaging principle problem of stochastic McKean-Vlasov equations with slow and fast time-scale is considered. Firstly, existence and uniqueness of the strong solutions of stochastic McKean-Vlasov equations with two time-scale is proved by using the Picard iteration. Secondly, we show that there exists an exponential convergence to...

This paper is devoted to the estimation of partial derivatives of multivariate density functions. In this regard, nonparametric linear wavelet-based estimators are introduced, showing their attractive properties from the theoretical point of view. In particular, we prove the strong uniform consistency properties of these estimators, over compact su...

This paper will prove the normal deviation of the synchronization of stochastic coupled system. According to the relationship between the stationary solution and the general solution, the martingale method is used to prove the normal deviation of the fixed initial value of the multi-scale system, thereby obtaining the normal deviation of the statio...

In this paper, a finite capacity two heterogeneous servers’ queuing system with retention of reneging customers is studied. The explicit transient probabilities of system size are obtained using matrix method. Further, the time-dependent mean and variance are presented. Finally, a numerical example is provided to show the behavior of the system.

In this paper, we shall prove a stochastic averaging principle for two-time-scale jump-diffusion SDEs under the non-Lipschitz coefficients.

An M/M/1 queueing system subjected to multiple differentiated vacations, customer impatience and a waiting server is analyzed. The explicit transient probabilities of system size are derived using probability generating function technique, Laplace transform, continued fractions and some properties of confluent hypergeometric function. Further, the...

In this paper we prove the stochastic homeomorphism flows for stochastic differential equations (SDEs) with non-Lipschitz coefficients and singular time.

Impatience behavior of an M/M/1 queueing system, which has differentiated server vacations, is considered. Arrivals follow a Poisson process, service is exponentially distributed, as are both vacation types. When the system is empty after waiting for a random period of time, the server takes a vacation and returns after a random duration. If there...

The existence and uniqueness theorem of solutions provides an effective tool for the model validation of both deterministic and stochastic equations. The objective of this paper is to establish the existence and uniqueness of solutions for a class of Itô-Doob stochastic fractional differential equations under non-Lipschitz condition which is weaker...

This paper deals with averaging principle for two-time-scale stochastic differential equations (SDEs) with non-Lipschitz coefficients, which extends the existing results: from Lipschitz to non-Lipschitz case. Under suitable conditions, the existence of an averaging equation eliminating the fast variable for coupled system is established, and as a r...

An M / M / 1 queue with reneging, catastrophes, server failures and repairs is considered. The arrivals follow a Poisson process and the servers serve according to an exponential distribution. On arrival a customer decides to join the queue and after joining the queue if a customer has to wait for the service longer than his expectation, he may ren...

In this paper, we deal with the averaging principle for a two-time-scale system of jump-diffusion stochastic differential equations. Under suitable conditions, we expand the weak error in powers of timescale parameter. We prove that the rate of weak convergence to the averaged dynamics is of order 1. This reveals that the rate of weak convergence i...

This work is devoted to averaging principle of a two-time-scale stochastic partial differential equation on a bounded interval $[0, l]$, where both the fast and slow components are directly perturbed by additive noises. Under some regular conditions on drift coefficients, it is proved that the rate of weak convergence for the slow variable to the a...

The synchronization of stochastic differential equations with both additive noise and linear multiplicative noise is investigated in pathwise sense, which generalize the results of [5] and [6]. In our situation, we can deal with the synchronization of the systems with mixed type noise, where one system has the additive noise, another has the linear...

The synchronization of stochastic differential equations (SDEs) driven by symmetric α-stable process and Brownian Motion is investigated in pathwise sense. This coupled dynamical system is a new mathematical model, where one of the systems is driven by Gaussian noise, another one is driven by non-Gaussian noise. In this paper, we prove that the syn...

Wong–Zakai type approximation for stochastic partial differential equations (abbreviate as PDEs) is well studied. Besides the polygonal approximation, a type of smooth noise approximation is considered. After showing the existence of random attractor for a class of random partial differential equations defined on the entire space (Formula presented...

Corresponding author at: School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, Hubei 430074, PR China.

The synchronization of stochastic differential equations (SDEs) with additive noise is investigated in pathwise sense, moreover convergence rate of synchronization is obtained. The optimality of the convergence rate is illustrated through examples.

This article deals with the weak error for averaging principle for two-time-scale system of jump-diffusion stochastic differential equation. Under suitable conditions, it is proved that the rate of weak convergence to the averaged effective dynamics is of order $1$ via an asymptotic expansion approach.

This article deals with the weak errors for averaging principle for a stochastic wave equation in a bounded interval $[0,L]$, perturbed by a oscillating term arising as the solution of a stochastic reaction-diffusion equation evolving with respect to the fast time. Under suitable conditions, it is proved that the rate of weak convergence to the ave...

This article deals with the weak errors for averaging principle for a stochastic wave equation in a bounded interval $[0,L]$, perturbed by a oscillating term arising as the solution of a stochastic reaction-diffusion equation evolving with respect to the fast time. Under suitable conditions, it is proved that the rate of weak convergence to the ave...

In this paper, we establish some conditional versions of the first part of the Borel-Cantelli lemma. As its applications, we study strong limit results of F-independent random variables sequences, the convergence of sums of F-independent random variables and the conditional version of strong limit results of the concomitants of order statistics.

In this article, we investigate averaging principle for stochastic hyperbolic-parabolic equations with two time-scales, in which both the slow and fast components are perturbed by multiplicative noises. Particularly, we prove that the rate of strong convergence for the slow component to the averaged dynamics is of order $1/2$, which significantly i...

In this paper, we study an averaging principle for stochastic FitzHugh-Nagumo system with different time scales driven by cylindrical Wiener processes and Poisson jumps, where the slow equation is non-autonomous and the fast equation is autonomous case. Under suitable assumptions, we show that the slow component mean-square strongly converges to th...

In this paper, we attempt to introduce a new numerical approach to solve backward doubly stochastic differential delay equation ( shortly-BDSDDEs ). In the beginning, we present some assumptions to get the numerical scheme for BDSDDEs, from which we prove important theorem. We use the relationship between backward doubly stochastic differential del...

In this paper, we present some assumptions to get the numerical scheme for backward doubly stochastic dierential delay equations (shortly-BDSDDEs), and we propose a scheme of BDSDDEs and discuss the numerical convergence and rate of convergence of our scheme.

In this note, we establish a strong convergence rate in averaging principle for stochastic FitzHugh–Nagumo system with two time-scales under weaker conditions.

We investigate the effect of domain perturbation on the behavior of mild solutions for a class of semilinear stochastic partial differential equations subject to the Dirichlet boundary condition. Under some assumptions, we obtain an estimate for the mild solutions under changes of the domain.

In this paper, we investigate the quasi sure large deviation for increments of a fractional Brownian sheet with Hurst index in Hölder norm on the rectangles. The quasi sure set of limit points for increments of a fractional Brownian sheet is also established.

This work concerns the problem associated with an averaging principle for two-time-scales stochastic partial differential equations (SPDEs) driven by cylindrical Wiener processes and Poisson random measures. Under suitable dissipativity conditions, the existence of an averaging equation eliminating the fast variable for the coupled system is proved...

In this paper, we first prove Schilder’s theorem in Hölder norm (0 ≤ α < 1) with respect to C r,p -capacity. Then, based on this result, we further prove a sharpening of large deviation principle for increments of fractional Brownian motion for C r,p -capacity in the stronger topology.

This article deals with averaging principle for stochastic hyperbolic-parabolic equations with slow and fast time-scales. Under suitable conditions, the existence of an averaging equation eliminating the fast variable for this coupled system is proved. As a consequence, an effective dynamics for slow variable which takes the form of stochastic wave...

The averaging principle for multivalued stochastic differential equations (MSDEs) driven by Brownian motion with Brownian noise is investigated. An averaged MSDEs for the original MSDEs is proposed, and their solutions are quantitatively compared. Under suitable assumptions, it is shown that the solution of the MSDEs converges to that of the origin...

This article deals with averaging principle for stochastic FitzHugh-Nagumo system with different time-scales. Under suitable conditions, the existence of an averaging equation eliminating the fast variable for this coupled system is proved, and as a consequence, the system can be reduced to a single stochastic ordinary equation with a modified coef...

We prove that Euler’s approximations for stochastic differential equations driven by infinite many Brownian motions and with non-Lipschitz coefficients converge almost surely. Moreover, the rate of convergence is obtained.

The existence and uniqueness of the solutions for a class of hyperbolic type stochastic evolution equations driven by some non-Gaussian Lévy processes are obtained. Moreover, an energy equality for the solutions of the equations is established. As examples, theses results are applied to a couple of stochastic wave type equations with jumps.

We prove some conditional Borel–Cantelli lemmas for sequences of random variables. As an application, a conditional version of the weighted Borel–Cantelli lemma is obtained.

In this paper, we first study the existence and uniqueness of solutions to the stochastic differential equations driven by fractional Brownian motion with non-Lipschitz coefficients. Then we investigate the explosion time in stochastic differential equations driven by fractional Browmian motion with respect to Hurst parameter more than half with sm...

In the note, we provide a method of finding conditional expectations based on its definition. As applications, two examples are provided, for which our method is simpler than that of Brzezniak-Zastawnia.

Two bilateral inequalities based on the Borel–Cantelli lemma and a non-negative sequence of bounded random variables were respectively obtained by Xie (2008, 2009). However, we observe that the upper bounds in the above cited references are greater than or equal to 1, so the upper bounds of these bilateral inequalities always hold true. In this n...

The theory of stochastic averaging principle provides an effective approach for the qualitative analysis of stochastic systems with different time-scales and is relatively mature for stochastic ordinary differential equations. In this paper, we study the averaging principle for a class of stochastic partial differential equations with two separated...

The aim of this paper is to consider the n-dimensional SDE: X t i =x i +∑ j=1 ∞ ∫ 0 t σ s ij (X s )dB s j +∫ 0 t b s i (X s )ds,i=1,2,⋯,n,(*) where {B t j } j=1 ∞ is an infinite sequence of independent standard Brownian motions. In this paper, we prove that the uniqueness in law for (*) implies the uniqueness of the joint distribution of a pair (X,...

Dynamical systems driven by Gaussian noises have been considered extensively in modeling, simulation and theory. However, complex systems in engineering and science are often subject to non-Gaussian fluctuations or uncertainties. A coupled dynamical system under non-Gaussian Lévy noise is considered. After discussing cocycle property, stationary or...

Dynamical systems driven by Gaussian noises have been considered extensively in modeling, simulation, and theory. However, complex systems in engineering and science are often subject to non-Gaussian fluctuations or uncertainties. A coupled dynamical system under a class of Lévy noises is considered. After discussing cocycle property, stationary or...

In this paper, we prove a sharpening of large deviation for increments of Brownian motion in (p,r)-capacity and Hölder norm case. As an application, we obtain a functional modulus of continuity for (p,r)-capacity in the stronger topology.

We prove large deviation principles for solutions of small perturbations of SDEs in Hölder norms and Sobolev norms, where the SDEs have non-Markovian coefficients. As an application, we obtain a large deviation principle for solutions of anticipating SDEs in terms of (r,p) capacities on the Wiener space.

In this paper, we prove that the process of product variation of a two–parameter smooth martingale admits an ∞ modification, which can be constructed as the quasi–sure limit of sum of the corresponding product variation.

We prove that, as Wiener functionals, local times of two-parameter martingales belong to some fractional Sobolev spaces over the Wiener space and, therefore, exist $(2,\infty )$-quasi surely.

Let X be a two parameter smooth semimartingale and ˜X be its process of the product variation. It is proved that ˜X can be approximated as D1−limit of sums of its discrete product variations as the mesh of division tends to zero. Moreover, this result can be strengthen to yield the quasi sure convergence of sums by estimating the speed of the conve...

Comparison theorems for solutions of one-dimensional backward stochastic differential equations were established by Peng and Cao-Yan, where the coefficients were, respectively, required to be Lipschitz and Dini continuous. In this work, we generalize the comparison theorem to the case where the coefficient is only continuous.

Let be a two-parameter semimartingale, where M is a continuous martingale, [Lambda] is the character of the Poisson point measure Y, N=Y-[Lambda], we prove that f(Xz) is expressible as such a sum once again via the partial differentiation formula, where f is a twice continuously differentiable function. Then, we prove a new theorem on the existence...