arXiv subjects

Xuerong Mao

Publications and source records attributed to Xuerong Mao.

At least 19 recordsLinked to original sources

arXiv2026

An explicit finite-memory scheme for approximating and sampling invariant measures of stochastic functional differential equations with infinite delay

Efficient sampling and numerical approximation of invariant probability measures (IPMs) on infinite-dimensional function spaces are important problems in scientific computing. In this paper, we study the numerical approximation and sampling of IPMs associated with stochastic functional differential equations with infinite delay (SFDEswID). To this end, we develop a fully explicit ergodicity-preserving truncated Euler--Maruyama scheme for SFDEswID that requires only finite historical storage and accommodates superlinearly growing coefficients. We establish strong convergence of the numerical segment process and show that it admits a unique IPM and is exponentially ergodic in the Wasserstein distance. Building on these results, we prove the convergence of the numerical IPM to the exact one and derive an explicit convergence rate. As a consequence, we obtain a quantitative long-time sampling error estimate of order $O\left(e^{-λ_\varepsilon t_n}+Δ^{ρ_\varepsilon}\right)$. The results provide a rigorous and computationally efficient framework for sampling IPMs and quantifying long-time sampling errors for stochastic systems with infinite delay.

math.NA↗

arXiv2025

Development of numerical methods for nonlinear hybrid stochastic functional differential equations with infinite delay

This paper addresses the challenging numerical simulation of nonlinear hybrid stochastic functional differential equations with infinite delays. We first propose an explicit scheme using space and time truncation, requiring only finite historical storage. Leveraging approximation theory, we prove the boundedness of the numerical solution's $p$th moment and establish its convergence, achieving a rate of $1/2$ order under polynomially growing coefficients. Furthermore, we refine the scheme to better capture the underlying exponential stability of the exact solution, in both moment and almost sure senses. Finally, numerical experiments are presented to validate our theoretical results.

math.NA↗

arXiv2025

Hybrid Stochastic Functional Differential Equations with Infinite Delay: Approximations and Numerics

This paper is to investigate if the solution of a hybrid stochastic functional differential equation (SFDE) with infinite delay can be approximated by the solution of the corresponding hybrid SFDE with finite delay. A positive result is established for a large class of highly nonlinear hybrid SFDEs with infinite delay. Our new theory makes it possible to numerically approximate the solution of the hybrid SFDE with infinite delay, via the numerical solution of the corresponding hybrid SFDE with finite delay.

math.PR↗

arXiv2025

Strong convergence of multiscale truncated Euler-Maruyama method for super-linear slow-fast stochastic differential equations

This manuscript is dedicated to the numerical approximation of super-linear slow-fast stochastic differential equations (SFSDEs). Borrowing the heterogeneous multiscale idea, we propose an explicit multiscale Euler-Maruyama scheme suitable for SFSDEs with locally Lipschitz coefficients using an appropriate truncation technique. By the averaging principle, we establish the strong convergence of the numerical solutions to the exact solutions in the pth moment. Additionally, under lenient conditions on the coefficients, we also furnish a strong error estimate. In conclusion, we give two illustrative examples and accompanying numerical simulations to affirm the theoretical outcomes.

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arXiv2025

Explicit positivity preserving numerical method for linear stochastic volatility models driven by $α$-stable process

In this paper, we introduce a linear stochastic volatility model driven by $α$-stable processes, which admits a unique positive solution. To preserve positivity, we modify the classical forward Euler-Maruyama scheme and analyze its numerical properties. The scheme achieves a strong convergence order of $1/α$. Numerical simulations are presented at the end to verify theoretical results.

math.PR↗

arXiv2024

Positivity-preserving truncated Euler and Milstein methods for financial SDEs with super-linear coefficients

In this paper, we propose two variants of the positivity-preserving schemes, namely the truncated Euler-Maruyama (EM) method and the truncated Milstein scheme, applied to stochastic differential equations (SDEs) with positive solutions and super-linear coefficients. Under some regularity and integrability assumptions we derive the optimal strong convergence rates of the two schemes. Moreover, we demonstrate flexibility of our approaches by applying the truncated methods to approximate SDEs with super-linear coefficients (3/2 and Aiıt-Sahalia models) directly and also with sub-linear coefficients (CIR model) indirectly. Numerical experiments are provided to verify the effectiveness of the theoretical results.

math.NA↗

arXiv2023

Invariance principles for G-brownian-motion-driven stochastic differential equations and their applications to G-stochastic control

The G-Brownian-motion-driven stochastic differential equations (G-SDEs) as well as the G-expectation, which were seminally proposed by Peng and his colleagues, have been extensively applied to describing a particular kind of uncertainty arising in real-world systems modeling. Mathematically depicting long-time and limit behaviors of the solution produced by G-SDEs is beneficial to understanding the mechanisms of system's evolution. Here, we develop a new G-semimartingale convergence theorem and further establish a new invariance principle for investigating the long-time behaviors emergent in G-SDEs. We also validate the uniqueness and the global existence of the solution of G-SDEs whose vector fields are only locally Lipchitzian with a linear upper bound. To demonstrate the broad applicability of our analytically established results, we investigate its application to achieving G-stochastic control in a few representative dynamical systems.

math.PR↗

arXiv2023

Wolbachia invasion to wild mosquito population in stochastic environment

Releasing sterile Wolbachia-infected mosquitoes to invade wild mosquito population is a method of mosquito control. In this paper, a stochastic mosquito population model with Wolbachia invasion perturbed by environmental fluctuation is studied. Firstly, well-posedness, positivity and Markov-Feller property of solution for this model are proved. Then a group of sharp threshold-type conditions is provided to characterize the long-term behavior of the model, which pinpoints the almost necessary and sufficient conditions for persistence and extinction of Wolbachia-infected and uninfected mosquito populations. Especially, our results indicates that even the initial Wolbachia infection frequency is low, the Wolbachia invasion into wild mosquito population can be promoted by stochastic environmental fluctuations. Finally, some numerical experiments are carried out to support our theoretical results.

math.DS↗

arXiv2022

Dynamics of an HIV/AIDS transmission model with protection awareness and fluctuations

We establish a stochastic HIV/AIDS model for the individuals with protection awareness and reveal how the protection awareness plays its important role in the control of AIDS. We firstly show that there exists a global positive solution for the stochastic model. By constructing Lyapunov functions, the ergodic stationary distribution when $R_{0}^{s}>1$ and the extinction when $R_{0}^{e}<1$ for the stochastic model are obtained. A number of numerical simulations by using positive preserving truncated Euler-Maruyama method (PPTEM) are performed to illustrate the theoretical results. Our new results show that the detailed publicity has great impact on the control of AIDS compared with the extensive publicity, while the continuous antiretroviral therapy (ART) is helpful in the control of HIV/AIDS.

math.DS↗

arXiv2022

An explicit approximation for super-linear stochastic functional differential equations

Since it is difficult to implement implicit schemes on the infinite-dimensional space, we aim to develop the explicit numerical method for approximating super-linear stochastic functional differential equations (SFDEs). Precisely, borrowing the truncation idea and linear interpolation we propose an explicit truncated Euler-Maruyama scheme for super-linear SFDEs, and obtain the boundedness and convergence in L^p. We also yield the convergence rate with 1/2 order. Different from some previous works, we release the global Lipschitz restriction on the diffusion coefficient. Furthermore, we reveal that numerical solutions preserve the underlying exponential stability. Moreover, we give several examples to support our theory.

math.NA↗

arXiv2022

The backward Euler-Maruyama method for invariant measures of stochastic differential equations with super-linear coefficients

The backward Euler-Maruyama (BEM) method is employed to approximate the invariant measure of stochastic differential equations, where both the drift and the diffusion coefficient are allowed to grow super-linearly. The existence and uniqueness of the invariant measure of the numerical solution generated by the BEM method are proved and the convergence of the numerical invariant measure to the underlying one is shown. Simulations are provided to illustrate the theoretical results and demonstrate the application of our results in the area of system control.

math.PR↗

arXiv2022

On the Analysis of a Generalised Rough Ait-Sahalia Interest Rate Model

Fractional Brownian motion with the Hurst parameter $H<\frac{1}{2}$ is used widely, for instance, to describe a 'rough' stochastic volatility process in finance. In this paper, we examine an Ait-Sahalia-type interest rate model driven by a fractional Brownian motion with $H<\frac{1}{2}$ and establish theoretical properties such as an existence-and-uniqueness theorem, regularity in the sense of Malliavin differentiability and higher moments of the strong solutions.

math.PR↗

arXiv2021

The truncated EM method for stochastic differential delay equations with variable delay

This paper mainly investigates the strong convergence and stability of the truncated Euler-Maruyama (EM) method for stochastic differential delay equations with variable delay whose coefficients can be growing super-linearly. By constructing appropriate truncated functions to control the super-linear growth of the original coefficients, we present a type of the truncated EM method for such SDDEs with variable delay, which is proposed to be approximated by the value taken at the nearest grid points on the left of the delayed argument. The strong convergence result (without order) of the method is established under the local Lipschitz plus generalized Khasminskii-type conditions and the optimal strong convergence order $1/2$ can be obtained if the global monotonicity with U function and polynomial growth conditions are added to the assumptions. Moreover, the partially truncated EM method is proved to preserve the mean-square and H_\infty stabilities of the true solutions. Compared with the known results on the truncated EM method for SDDEs, a better order of strong convergence is obtained under more relaxing conditions on the coefficients, and more refined technical estimates are developed so as to overcome the challenges arising due to variable delay. Lastly, some numerical examples are utilized to confirm the effectiveness of the theoretical results.

math.NA↗

arXiv2020

Delay Feedback Control for Switching Diffusion Systems Based on Discrete Time Observations

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 observations of the system states and the Markovian states in order for the controlled switching diffusion system (SDS) to be exponentially stable in $p$th moment and probability one as well as stable in $H_\infty$. The designed control principles are implementable to stablize quasi-linear and highly nonlinear SDSs. For quasi-linear SDSs the criteria are sharp that under the control with high strength the controlled SDSs will be stable (bounded) while under the weaker control they will be unstable (unbounded) in mean square. The sample and moment Lyapunov exponents are estimated which have close relationship with the time delays.

math.OC↗

arXiv2020

Delay-dependent Asymptotic Stability of Highly Nonlinear Stochastic Differential Delay Equations Driven by $G$-Brownian Motion

Based on the classical probability, the stability criteria for stochastic differential delay equations (SDDEs) where their coefficients are either linear or nonlinear but bounded by linear functions have been investigated intensively. Moreover, the dependent stability of the highly nonlinear hybrid stochastic differential equations is recently studied. In this paper, by using the nonlinear expectation theory, we explore the dependent stability of a class of highly nonlinear hybrid stochastic differential delay equations driven by $G$-Brownian motion ($G$-SDDEs). Firstly, we give preliminaries of sublinear expectation. Then, the delay-dependent criteria of the stability and boundedness of solutions to $G$-SDDEs is provided. Finally, an illustrative example is analyzed by the $φ$-max-mean algorithm.

math.OC↗

arXiv2020

Strong convergence and asymptotic stability of explicit numerical schemes for nonlinear stochastic differential equations

In this article we introduce several kinds of easily implementable explicit schemes, which are amenable to Khasminski's techniques and are particularly suitable for highly nonlinear stochastic differential equations (SDEs). We show that without additional restriction conditions except those which guarantee the exact solutions possess their boundedness in expectation with respect to certain Lyapunov functions, the numerical solutions converge strongly to the exact solutions in finite-time. Moreover, based on the nonnegative semimartingale convergence theorem, positive results about the ability of explicit numerical approximation to reproduce the well-known LaSalle-type theorem of SDEs are proved here, from which we deduce the asymptotic stability of numerical solutions. Some examples and simulations are provided to support the theoretical results and to demonstrate the validity of the approach.

math.NA↗

arXiv2019

Advances in Stabilisation of Hybrid Stochastic Differential Equations by Delay Feedback Control

A novel approach to design the feedback control based on past states is proposed for hybrid stochastic differential equations (HSDEs). This new theorem builds up the connection between the delay feedback control and the control function without delay terms, which enables one to construct the delay feedback control using the existing results on stabilities of HSDEs. Methods to find the upper bound of the length of the time delay are also investigated. Numerical simulations are presented to demonstrate the new theorem.

math.OC↗

arXiv2019

Truncated Euler-Maruyama method for classical and time-changed non-autonomous stochastic differential equations

The truncated Euler-Maruyama (EM) method is proposed to approximate a class of non-autonomous stochastic differential equations (SDEs) with the Hölder continuity in the temporal variable and the super-linear growth in the state variable. The strong convergence with the convergence rate is proved. Moreover, the strong convergence of the truncated EM method for a class of highly non-linear time-changed SDEs is studied.

math.NA↗