Chenggui Yuan

• Swansea University

In this paper, the discrete parametrix method is adopted to investigate the estimation of transition kernel for Euler-Maruyama scheme SDEs driven by α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemar...

In this paper, we study small-time asymptotic behaviors for a class of distribution dependent stochastic differential equations driven by fractional Brownian motions with Hurst parameter $H\in(1/2,1)$ and magnitude $\ep^H$. By building up a variational framework and two weak convergence criteria in the factional Brownian motion setting, we establis...

In this paper, we investigate the weak convergence rate of Euler-Maruyama’s approximation for stochastic differential equations with low regular drifts. Explicit weak convergence rates are presented if drifts satisfy an integrability condition including discontinuous functions which can be non-piecewise continuous or in some fractional Sobolev spac...

In this paper we study a class of distribution dependent stochastic differential equations driven by fractional Brownian motions with Hurst parameter H∈(0,1/2)∪(1/2,1). We prove the well-posedness of this type equations, and then establish a general result on the Bismut formula for the Lions derivative by using Malliavin calculus. As applications,...

In this paper, the quadratic transportation cost inequality for SDEs with Dini continuous drift and the W1-transportation cost inequality for SDEs with singular coefficients are established via the stability of the Wasserstein distance and relative entropy of measures under the homeomorphism induced by Zvonkin’s transformation.

Under a Lipschitz condition on distribution dependent coefficients, the central limit theorem and the moderate deviation principle are obtained for solutions of McKean-Vlasov type stochastic differential equations, which generalize the corresponding results for classical stochastic differential equations to the distribution dependent setting.

We show existence of an invariant probability measure for a class of functional McKean-Vlasov SDEs by applying Kakutani's fixed point theorem to a suitable class of probability measures on a space of continuous functions. Unlike some previous works, we do not assume a monotonicity condition to hold. Further, our conditions are even weaker than some...

In this paper we study a class of distribution dependent stochastic differential equations driven by fractional Brownian motions with Hurst parameter H\in(1/2,1). We prove the well-posedness of this type equations, and then establish a general result on the Bismut formula for the Lions derivative by using Malliavin calculus. As applications, we pro...

In this paper, the weak Harris theorem developed in [18] is illustrated by using a straightforward Wasserstein coupling, which implies the exponential ergodicity of the functional solutions to a range of neutral type SDEs with infinite length of memory. A concrete example is presented to illustrate the main result.

For the sake of saving time and costs the feedback control based on discrete-time observations is used to stabilize the switching diffusion systems. Response lags are required by most of physical systems and play a key role in the feedback control. The aim of this paper is to design delay feedback control functions based on the discrete-time observ...

This work is concerned with the stability of regime-switching processes under the perturbation of the transition rate matrices From the viewpoint of application, two kinds of perturbations are studied: the size of the transition rate matrix is fixed, and only the values of entries are perturbed; the values of entries and the size of the transition...

In this paper, we are concerned with convergence rate of Euler–Maruyama scheme for stochastic differential equations with Hölder–Dini continuous drifts. The key contributions are as follows: (i) by means of regularity of non-degenerate Kolmogrov equation, we investigate convergence rate of Euler–Maruyama scheme for a class of stochastic differentia...

In this paper, a multilevel Monte Carlo theta EM scheme is provided for stochastic differential delay equations with small noise. Under a global Lipschitz condition, the variance of two coupled paths is derived. Then, the global Lipschitz condition is replaced by one-sided Lipschitz condition, in order to guarantee the moment finiteness of numerica...

By using limit theorems of uniform mixing Markov processes and martingale difference sequences, the strong law of large numbers, central limit theorem, and the law of iterated logarithm are established for additive functionals of path-dependent stochastic differential equations.

In this paper, the existence and uniqueness of strong solutions to distribution dependent neutral SFDEs are proved. We give the conditions such that the order preservation of these equations holds. Moreover, we show these conditions are also necessary in the delay case of the neutral term.

This paper investigates dynamic behaviors of the tumor-immune system perturbed by environmental noise. The model describes the response of the cytotoxic T lymphocyte (CTL) to the growth of an immunogenic tumour. The main methods are stochastic Lyapunov analysis, comparison theorem for stochastic differential equations (SDEs) and strong ergodicity t...

In this paper, some new results on the the regularity of Kolmogorov equations associated to the infinite dimensional OU-process are obtained. As an application, the average L²-error on [0, T] of exponential integrator scheme for a range of semi-linear stochastic partial differential equations is derived, where the drift term is assumed to be Hölder...

This work is concerned with the stability of regime-switching processes under the perturbation of the transition rate matrices. From the viewpoint of application, two kinds of perturbations are studied: the size of the transition rate matrix is fixed, and only the values of entries are perturbed; the values of entries and the size of the transition...

By using the Euler-Maruyama approximation and a localization argument, the existence and uniqueness are addressed for a class of neutral type SDEs with infinite length of memory. Moreover, the weak Harris theorem developed in \cite{HMS11} is illustrated by using a straightforward Wasserstein coupling, which implies the exponential ergodicity of the...

By using the Euler-Maruyama approximation and a localization argument, the existence and uniqueness are addressed for a class of neutral type SDEs with infinite length of memory. Moreover, the weak Harris theorem developed in [18] is illustrated by using a straightforward Wasserstein coupling, which implies the exponential ergodicity of the functio...

The existence and uniqueness of the numerical invariant measure of the backward Euler-Maruyama method for stochastic differential equations with Markovian switching is yielded, and it is revealed that the numerical invariant measure converges to the underlying invariant measure in the Wasserstein metric. Under the polynomial growth condition of dri...

This paper is concerned with strong convergence of the truncated Euler-Maruyama scheme for neutral stochastic differential delay equations driven by Brownian motion and pure jumps respectively. Under local Lipschitz condition, convergence rates of the truncated EM scheme are given.

Under a local one-sided Lipschitz condition, Krylov [KR] proved the existence and uniqueness of the strong solutions for stochastic differential equations by using the Euler-Maruyama approximation, where he showed that the sequence of numerical solutions converges to the true solution in probability as the stepsize tends to zero. In this note, we s...

The asymptotic log-Harnack inequality is established for several different models of stochastic differential systems with infinite memory: non-degenerate SDEs, Neutral SDEs, semi-linear SPDEs, and stochastic Hamiltonian systems. As applications, the following properties are derived for the associated segment Markov semigroups: asymptotic heat kerne...

This article focuses on a class of two-time-scale functional stochastic differential equations, where the phase space of the segment processes is infinite-dimensional. The systems under consideration have a fast-varying component and a slowly varying one. First, the ergodicity of the fast-varying component is obtained. Then inspired by the Khasmins...

In this work, we are concerned with existence and uniqueness of invariant measures for path-dependent random diffusions and their time discretizations. The random diffusion here means a diffusion process living in a random environment characterized by a continuous time Markov chain. Under certain ergodic conditions, we show that the path-dependent...

In this paper, exploiting the regularities of the corresponding Kolmogorov equations involved we investigate strong convergence of exponential integrator scheme for a range of stochastic partial differential equations, in which the drift term is H\"older continuous, and reveal the rate of convergence.

This paper is concerned with strong convergence and almost sure convergence for neutral stochastic differential delay equations under non-globally Lipschitz continuous coefficients. Convergence rates of $\theta$-EM schemes are given for these equations driven by Brownian motion and pure jumps respectively, where the drift terms satisfy locally one-...

This paper is concerned with strong convergence of a tamed $\theta$-Euler-Maruyama scheme for neutral stochastic differential delay equations with superlinearly growing coefficients. We not only prove the strong convergence of implicit schemes, but also reveal the convergence rate for these equations driven by Brownian motion and pure jumps, respec...

In this chapter, we investigate ergodicity for certain classes of functional stochastic equations including functional stochastic differential equations (FSDEs for short) driven by Brownian motions, FSDEs of neutral type, FSDEs driven by jump processes, functional stochastic partial differential equations (FSPDEs for abbreviation) driven by cylindr...

In this chapter , we investigate convergence rate of the Euler–Maruyama (EM) scheme for a class of SDDEs, where the corresponding coefficients may be highly nonlinear w.r.t. the delay variables. More precisely, we reveal that convergence rate of the EM scheme is \(\frac{1}{2}\) for the Brownian motion case. As for EM approximation for the pure jump...

In this chapter, we derive the convergence of the long-term return \(t^{-\mu }\int _0^tX(s)\text {d}s\) for some \(\mu \ge 1\), where X is the short-term interest rate that is an extension of the Cox-Ingersoll-Ross model with jumps and memory. We also investigate the corresponding behavior of the two-factor Cox-Ingersoll-Ross model with jumps and m...

In this chapter, by the weak convergence method , based on a variational representation for positive functionals of a Poisson random measure and a Brownian motion, we establish uniform large deviation principles (LDPs for short) for a class of FSDEs of neutral type driven by jump processes.KeywordsUniform Large DeviationPoisson Random MeasureNeutra...

Dealing with diffusions with “pure delay” in which both the drift and the diffusion coefficients depend only on the arguments with delays, most of the existing results are not applicable. This chapter uses variation-of-constants formulae to overcome the difficulties due to the lack of the information at the current time, and establishes existence a...

This paper focuses on stochastic partial differential equations (SPDEs) under two-time-scale formulation. Distinct from the work in the existing literature, the systems are driven by $\alpha$-stable processes with $\alpha \in(1,2)$. In addition, the SPDEs are either modulated by a continuous-time Markov chain with a finite state space or have an ad...

In this paper, we are concerned with convergence rate of Euler-Maruyama scheme for stochastic differential equations with rough coefficients. The key contributions lie in (i), by means of regularity of non-degenerate Kolmogrov equation, we investigate convergence rate of Euler-Maruyama scheme for a class of stochastic differential equations, which...

In this paper, we are concerned with long-time behavior of Euler-Maruyama schemes associated with a range of regime-switching diffusion processes. The key contributions of this paper lie in that existence and uniqueness of numerical invariant measures are addressed (i) for regime-switching diffusion processes with finite state spaces by the Perron-...

In this paper, we investigate the convergence of the tamed Euler-Maruyama (EM) scheme for a class of neutral stochastic differential delay equations. The strong convergence results of the tamed EM scheme are presented under global and local non-Lipschitz conditions, respectively. Moreover, under a global Lipschitz condition, we provide the rate of...

Some classes of stochastic fractional differential equations with respect to time and a pseudo-differential equation with respect to space are investigated. Using estimates for the Mittag-Leffler functions and a fixed point theorem, existence and uniqueness of mild solutions of the equations under consideration are established.

This work is concerned with the approximate controllability of nonlinear fractional impulsive stochastic differential system under the assumption that the corresponding linear system is approximately controllable. Using fractional calculus, stochastic analysis, and the technique of stochastic control theory, a new set of sufficient conditions for t...

This work is concerned with the approximate controllability of a nonlinear fractional impulsive evolution system under the assumption that the corresponding linear system is approximate controllable. Using the fractional calculus, the Krasnoselskii fixed point theorem, and the technique of controllability theory, some new sufficient conditions for...

In this article, we study a class of stochastic parabolic equations by fractional noise with Markovian switching. Based on the explicit representation of the strong solution given by an evolution system, we investigate the pth moment and almost surely exponential stabilities with the exponential rate function t2H.

In this paper, we mainly investigate the convergence rate of the Euler-Maruyama(EM) scheme for two classes of neutral stochastic differential delay equations (NSDDEs), one is driven by Brownian motion, the other is driven by pure jump process. The coefficients in both classes of NSDDEs maybe highly non-linear with respect to the delay variables. Un...

This paper focuses on a class of two-time-scale functional stochastic differential equations, where the phase space of the segment processes is infinite-dimensional. It develops ergodicity of the fast component and obtains a strong limit theorem for the averaging principle in the spirit of Khasminskii's averaging approach for the slow component.

In this paper, we prove the existence and uniqueness of the solution for neutral stochastic differential delay equations with locally monotone coefficients by using numerical approximation. An example is provided to illustrate our theory.

By using the Malliavin calculus, we establish the transportation-cost inequalities for stochastic differential equations with jumps, which generalizes the results in [14]. As an application, we establish the transportation-cost inequalities for regime-switching diffusion processes.

Explicit sufficient conditions on the hypercontractivity are presented for two classes of functional stochastic partial differential equations driven by, respectively, non-degenerate and degenerate Gaussian noises. Consequently, these conditions imply that the associated Markov semigroup is $L^2$-compact and exponentially convergent to the stationa...

In this paper, we discuss hypercontractivity for the Markov semigroup $P_t$ which is generated by segment processes associated with a range of functional SDEs of neutral type. As applications, we also reveal that the semigroup $P_t$ converges exponentially to its unique invariant probability measure $\mu$ in entropy, $L^2 (\mu)$ and $\|\cdot\|_{\mb...

In this paper, we focus on stochastic reaction-diffusion equations with jumps. By a new auxiliary function, we investigate non-negative property of the local strong (variational) solutions, which applies to stochastic reaction-diffusion equations with highly nonlinear noise terms. As a byproduct, we study the problem of non-existence of global stro...

In this note, we discuss strong convergence of exponential integrator scheme based on spatial and time discretization for a class of neutral stochastic partial differential equations driven by α-stable processes.

In this paper, by the weak convergence method, based on a variational representation for positive functionals of a Poisson random measure and Brownian motion, we establish uniform large deviation principles (LDPs) for a class of neutral stochastic differential equations driven by jump processes. As a byproduct, we also obtain uniform LDPs for neutr...

In this paper, we discuss long-time behavior of sample paths for a wide range of regime-switching diffusions. Firstly, almost sure asymptotic stability is concerned (i) for regime-switching diffusions with finite state spaces by the Perron-Frobenius theorem, and, with regard to the case of reversible Markov chain, via the principal eigenvalue appro...

Exponential stability of the exact solutions as well as $\theta$-EM ($\frac{1}{2}<\theta\le 1$) approximations to neutral stochastic differential delay equations with Markov switching will be investigated in this paper. Sufficient conditions are obtained to ensure the $p$-th moment ($p\ge1$) and almost sure exponential stability of the exact soluti...

In this paper we discuss an exponential integrator scheme, based on spatial discretization and time discretization, for a class of stochastic partial differential equations. We show that the scheme has a unique stationary distribution whenever the step size is sufficiently small, and that the weak limit of the stationary distribution of the scheme...

An explicit sufficient condition on the hypercontractivity is derived for the Markov semigroup associated to a class of functional stochastic differential equations. Consequently, the semigroup $P_t$ converges exponentially to its unique invariant probability measure $\mu$ in both $L^2(\mu)$ and the totally variational norm, and it is compact in $L...

In this paper, using the remote start or dissipative method, we investigate ergodicity for several kinds of functional stochastic equations including functional stochastic differential equations (SDEs) with variable delays, neutral functional SDEs, functional SDEs driven by jump processes, and semi-linear functional stochastic partial differential...

Retarded stochastic differential equations (SDEs) constitute a large collection of systems arising in various real-life applications. Most of the existing results make crucial use of dissipative conditions. Dealing with "pure delay" systems in which both the drift and the diffusion coefficients depend only on the arguments with delays, the existing...

In this paper, we discuss exponential mixing property for Markovian semigroups generated by segment processes associated with several class of retarded Stochastic Differential Equations (SDEs) which cover SDEs with constant/variable/distributed time-lags. In particular, we investigate the exponential mixing property for (a) non-autonomous retarded...

Using Malliavin calculus, explicit derivative formulae are established for a class of semi-linear functional stochastic partial differential equations with additive or multiplicative noise. As applications, gradient estimates and Harnack inequalities are derived for the semigroup of the associated segment process.

In this paper, we show that the exponential integrator scheme both in spatial discretization and time discretization for a class of stochastic partial differential equations has a unique stationary distribution whenever the stepsize is sufficiently small, and reveal that the weak limit of the law for the exponential integrator scheme is in fact the...

In this paper, we consider a class of multi-dimensional reflected stochastic delay differential equations with jumps. Based on the existence and uniqueness of the strong solution to the equation, we prove that the Markov semigroup generated by the segment process corresponding to the solution admits a unique invariant measure on the Skorohod space...

This paper concerns Razumikhin-type theorems on exponential stability of stochastic differential delay equations with Markovian switching, where the modulating Markov chain involves small parameters. The smaller the parameter is, the rapider switching the system will experience. In order to reduce the complexity, we will “replace” the original syst...

By using coupling arguments, Harnack type inequalities are established for a class of stochastic (functional) differential equations with multiplicative noises and non-Lipschitzian coefficients. To construct the required couplings, two results on existence and uniqueness of solutions on an open domain are presented.

By using Girsanov transformation and martingale representation, Talagrand-type transportation cost inequalities, with respect to both the uniform and the $L^2$ distances on the global free path space, are established for the segment process associated to a class of neutral functional stochastic differential equations. Neutral functional stochastic...

This paper considers singular systems that involve both continuous dynamics and discrete events with the coefficients being modulated by a continuous-time Markov chain. The underlying systems have two distinct characteristics. First, the systems are singular, that is, characterized by a singular coefficient matrix. Second, the Markov chain of the m...

A class of functional differential equations are investigated. Using the Girsanov-transformation argument we establish the quadratic transportation cost inequalities for a class of finite-dimensional neutral functional stochastic differential equations and infinite-dimensional neutral functional stochastic partial differential equations under diffe...

It is with great pleasure that we dedicate this paper to Professor Rudolf Gorenflo. In particular, the second named author wants to express his appreciation to an outstanding mathematician who was to him more than 30 years ago an inspiring teacher, and a very humane senior colleague later on. A class of fractional differential equations are investi...

In this paper we investigate the convergence rate of the Euler-Maruyama (EM) scheme for a class of stochastic differential delay equations, where the corresponding coefficients may be highly nonlinear with respect to the delay variables. In particular, we reveal that the convergence rate of the Euler-Maruyama scheme is 12 for the Brownian motion ca...

The long-term interest rates, for example, determine when homeowners refinance their mortgages in mortgage pricing, play a dominant role in life insurance, decide when one should exchange a long bond to a short bond in pricing an option. In this paper, for a one-factor model, we reveal that the long-term return t-(mu) integral(t)(0) X (s)ds for som...

In this paper, we investigate a class of hybrid stochastic heat equations. By explicit formulae of solutions, we not only reveal the sample Lyapunov exponents but also discuss the $p$th moment Lyapnov exponents. Moreover, several examples are established to demonstrate that unstable (deterministic or stochastic) dynamical systems can be stabilized...

By using Malliavin calculus, explicit derivative formulae are established for a class of semi-linear functional stochastic partial differential equations with additive or multiplicative noise. As applications, gradient estimates and Harnack inequalities are derived for the semigroup of the associated segment process. Keywords: Bismut formula, Malli...

By constructing successful couplings, the derivative formula, gradient estimates and Harnack inequalities are established for the semigroup associated with a class of degenerate functional stochastic differential equations.

In this paper we focus on the pathwise stability of mild solutions for a class of stochastic partial differential equations which are driven by switching-diffusion processes with jumps. In comparison to the existing literature, we show that: (i) the criterion to guarantee pathwise stability does not rely on the moment stability of the system; (ii)...

In this paper, we are interested in the numerical solutions of stochastic functional differential equations (SFDEs) with {\it jumps}. Under the global Lipschitz condition, we show that the $p$th moment convergence of the Euler-Maruyama (EM) numerical solutions to SFDEs with jumps has order $1/p$ for any $p\ge 2$. This is significantly different fro...

This paper considers stochastic population dynamics driven by Levy noise. The contributions of this paper lie in that (a) Using Khasminskii-Mao theorem, we show that the stochastic differential equation associated with the model has a unique global positive solution; (b) Applying an exponential martingale inequality with jumps, we discuss the asymp...

We extend the stability criterion in distribution as in Yuan and Mao [C. Yuan and X. Mao. Asymptotic stability in distribution of stochastic differential equations with Markovian switching, Stochastic Process. Appl. 103 (2003) 277–291] to multi-dimensional reflected Markov-modulated stochastic differential equations on a closed positive orthant.

We focus in this paper on the stochastic stabilization problems of PDEs by Levy noise. Sufficient conditions under which the perturbed systems decay exponentially with a general rate function are provided and some examples are constructed to demonstrate the applications of our theory.

This paper considers singular systems that involve both continuous dynamics and discrete events with the coefficients being modulated by a continuous-time Markov chain. The underlying systems have two distinct characteristics. First, the systems are singular, that is, characterized by a singular coefficient matrix. Second, the Markov chain of the m...

This paper considers competitive Lotka-Volterra population dynamics with jumps. The contributions of this paper are as follows. (a) We show stochastic differential equation (SDE) with jumps associated with the model has a unique global positive solution; (b) We discuss the uniform boundedness of $p$th moment with $p>0$ and reveal the sample Lyapuno...

In this paper we establish a comparison theorem for stochastic differential delay equations with jumps. An example is constructed to demonstrate that the comparison theorem need not hold whenever the diffusion term contains a delay function although the jump-diffusion coefficient could contain a delay function. Moreover, another example is establis...

We investigate stochastic partial differential equations with jumps in infinite dimensions. The key motivation of this paper is, under a local Lipschitz condition but without a linear growth condition, to give an existence-and-uniqueness theorem (Khasminskii-type theorem) of which the classical existence-and-uniqueness result can be regarded as a s...

By constructing a new coupling, the log-Harnack inequality is established for the functional solution of a delay stochastic differential equation with multiplicative noise. As applications, the strong Feller property and heat kernel estimates w.r.t. quasi-invariant probability measures are derived for the associated transition semigroup of the solu...

This study investigates sufficient conditions for stability of delay jump diffusion processes in the sense of almost sure stability, stability in distribution, and exponential stability in mean square. The Lyapunov function method and the Razumikhin argument play an important role in this study.

The priority of Liming Wu for the Poincar\'e inequality for birth-death processes on a Poisson space is announced.

This work is concerned with stability of stochastic differential delay equations with Markovian switching, where the modulating Markov chain has a large state space and is subject to both fast and slow movements. Under simple conditions, we demonstrate that if the limit systems are pth-moment exponentially stable, then the original systems are pth-...

In this work, we investigate stochastic partial differential equations with variable delays and jumps. We derive by estimating the coefficients functions in the stochastic energy equality some sufficient conditions for exponential stability and almost sure exponential stability of energy solutions, and generalize the results obtained by Taniguchi [...

We investigate stochastic partial differential equations. By introducing a suitable metric between the transition probability functions of mild solutions, we derive sufficient conditions for stability in distribution of mild solutions. Consequently, we generalize some existing results to infinite dimensional cases. Finally, one example is construct...

In this paper we study the well-known Khasminskii-Type Theorem, i.e. the existence and uniqueness of solutions of stochastic evolution delay equations, under local Lipschitz condition, but without linear growth condition. We then establish one stochastic LaSalle-type theorem for asymptotic stability analysis of strong solutions. Moreover, several e...

In this paper, we investigate the existence and uniqueness of solutions to stochastic differential delay equations under a local Lipschitz condition but without linear growth condition on its coefficients. Moreover, we prove convergence in probability of the Euler–Maruyama approximation as well as of the stochastic theta method approximation to the...

We focus on stochastic delay differential equations in the form: for τ>0 d[x(t)-G(x(t-τ))]=f(x(t),x(t-τ),r(t))dt+g(x(t),x(t-τ),r(t))dB(t),t≥0,(2·1) We are concerned with neutral stochastic differential delay equations with Markovian switching (NSDDEwMSs). We derive sufficient conditions for stability in distribution and generalize some results of B...

In this article, we investigate the strong convergence of the Euler-Maruyama method and stochastic theta method for stochastic differential delay equations with jumps. Under a global Lipschitz condition, we not only prove the strong convergence, but also obtain the rate of convergence. We show strong convergence under a local Lipschitz condition an...

The existence, uniqueness and some sufficient conditions for stability in distribution of mild solutions to stochastic partial differential delay equations with jumps are presented. The principle technique of our investigation is to construct a proper approximating strong solution system and carry out a limiting type of argument to pass on stabilit...

Focusing on hybrid diffusions in which continuous dynamics and discrete events coexist, this work is concerned with approximation of solutions for hybrid stochastic differential equations with a state-dependent switching process. Iterative algorithms are developed. The continuous-state-dependent switching process presents added difficulties in anal...

Nonlinear differential equations have been used for decades for studying fluctuations in the populations of species, interactions of species with the environment, and competition and symbiosis between species. Over the years, the original non-linear models have been embellished with delay terms, stochastic terms and more recently discrete dynamics....