arXiv subjects

Xuerong Mao

Publications and source records attributed to Xuerong Mao.

36 records · Page 2Linked to original sources

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↗

arXiv2018

Generalized Ait-Sahalia-type interest rate model with Poisson jumps and convergence of the numerical approximation

In this paper, we consider the generalized Ait-Sahaliz interest rate model with Poisson jumps in finance. The analytical properties including the positivity, boundedness and pathwise asymptotic estimations of the solution to the model are investigated. Moreover, we prove that the Euler-Maruyama (EM) numerical solutions will converge to the true solution in probability. Finally, under assumption that the interest rate or the asset price is governed by this model, we apply the EM solutions to compute some financial quantities.

math.NA↗

arXiv2018

The truncated EM method for stochastic differential equations with Poisson jumps

In this paper, we use the truncated EM method to study the finite time strong convergence for the SDEs with Poisson jumps under the Khasminskii-type condition. We establish the finite time $ \mathcal L ^r (r \ge 2) $ convergence rate when the drift and diffusion coefficients satisfy super-linear condition and the jump coefficient satisfies the linear growth condition. The result shows that the optimal $\mathcal L ^r$-convergence rate is close to $ 1/ (1 + γ)$, where $γ$ is the super-linear growth constant. This is significantly different from the result on SDEs without jumps. When all the three coefficients of SDEs are allowing to grow super-linearly, the $ \mathcal L^r (0<r<2)$ strong convergence results are also investigated and the optimal strong convergence rate is shown to be not greater than $1/4$. Moreover, we prove that the truncated EM method preserve nicely the mean square exponentially stability and asymptotic boundedness of the underlying SDEs with Piosson jumps. Several examples are given to illustrate our results.

math.NA↗

arXiv2017

The truncated milstein method for stochastic differential equations

Inspired by the truncated Euler-Maruyama method developed in Mao (J. Comput. Appl. Math. 2015), we propose the truncated Milstein method in this paper. The strong convergence rate is proved to be close to 1 for a class of highly non-linear stochastic differential equations. Numerical examples are given to illustrate the theoretical results.

math.NA↗

arXiv2017

The Truncated Euler-Maruyama Method for Stochastic Differential Delay Equations

The numerical solutions of stochastic differential delay equations (SDDEs) under the generalized Khasminskii-type condition were discussed by Mao [15], and the theory there showed that the Euler-Maruyama (EM) numerical solutions converge to the true solutions in probability. However, there is so far no result on the strong convergence (namely in L^p) of the numerical solutions for the SDDEs under this generalized condition. In this paper, we will use the truncated EM method developed by Mao [16] to study the strong convergence of the numerical solutions for the SDDEs under the generalized Khasminskii-type condition.

math.NA↗

arXiv2017

Multi-level Monte Carlo methods with the Truncated Euler-Maruyama Scheme for Stochastic Differential Equations

In this paper, the truncated Euler-Maruyama (EM) method is employed together with the Multi-level Monte Carlo (MLMC) method to approximate the expectations of functions of solutions to stochastic differential equations (SDEs). The convergence rate and the computational cost of the approximations using the truncated EM method with the MLMC method are proved when the coefficients of SDEs fulfill the local Lipschitz and Khasminskii-type conditions. Numerical examples are given to demonstrate the theoretical results.

math.NA↗

arXiv2015

The Hitting Times of A Stochastic Epidemic Model

In this paper, we focus on the hitting times of a stochastic epidemic model presented by \cite{Gray}. Under the help of the auxiliary stopping times, we investigate the asymptotic limits of the hitting times by the variations of calculus and the large deviation inequalities when the noise is sufficiently small. It can be shown that the relative position between the initial state and the hitting state determines the scope of the hitting times greatly.

math.PR↗

arXiv2014

Mean Square Polynomial Stability of Numerical Solutions to a Class of Stochastic Differential Equations

The exponential stability of numerical methods to stochastic differential equations (SDEs) has been widely studied. In contrast, there are relatively few works on polynomial stability of numerical methods. In this letter, we address the question of reproducing the polynomial decay of a class of SDEs using the Euler--Maruyama method and the backward Euler--Maruyama method. The key technical contribution is based on various estimates involving the gamma function.

math.PR↗

arXiv2013

On the almost sure running maxima of solutions of affine neutral stochastic functional differential equations

This paper studies the large fluctuations of solutions of finite--dimensional affine stochastic neutral functional differential equations with finite memory, as well as related nonlinear equations. We find conditions under which the exact almost sure growth rate of the running maximum of each component of the system can be determined, both for affine and nonlinear equations. The proofs exploit the fact that an exponentially decaying fundamental solution of the underlying deterministic equation is sufficient to ensure that the solution of the affine equation converges to a stationary Gaussian process.

math.PR↗

arXiv2013

On the Existence and Uniqueness of Solutions of Stochastic Equations of Neutral Type

This paper considers some the existence and uniqueness of strong solutions of stochastic neutral functional differential equations. The conditions on the neutral functional relax those commonly used to establish the existence and uniqueness of solutions of NSFDEs for many important classes of functional, and parallel the conditions used to ensure existence and uniqueness of solutions of related deterministic neutral equations. Exponential estimates on the almost sure and p--th mean rate of growth of solutions under the weaker existence conditions are also given.

math.PR↗

arXiv2013

Stability Analysis of Continuous-Time Switched Systems with a Random Switching Signal

This paper is concerned with the stability analysis of continuous-time switched systems with a random switching signal. The switching signal manifests its characteristics with that the dwell time in each subsystem consists of a fixed part and a random part. The stochastic stability of such switched systems is studied using a Lyapunov approach. A necessary and sufficient condition is established in terms of linear matrix inequalities. The effect of the random switching signal on system stability is illustrated by a numerical example and the results coincide with our intuition.

eess.SY↗

arXiv2012

Strong convergence and stability of implicit numerical methods for stochastic differential equations with non-globally Lipschitz continuous coefficients

We are interested in the strong convergence and almost sure stability of Euler-Maruyama (EM) type approximations to the solutions of stochastic differential equations (SDEs) with non-linear and non-Lipschitzian coefficients. Motivation comes from finance and biology where many widely applied models do not satisfy the standard assumptions required for the strong convergence. In addition we examine the globally almost surely asymptotic stability in this non-linear setting for EM type schemes. In particular, we present a stochastic counterpart of the discrete LaSalle principle from which we deduce stability properties for numerical methods.

math.NA↗

arXiv2012

Convergence, Non-negativity and Stability of a New Milstein Scheme with Applications to Finance

We propose and analyse a new Milstein type scheme for simulating stochastic differential equations (SDEs) with highly nonlinear coefficients. Our work is motivated by the need to justify multi-level Monte Carlo simulations for mean-reverting financial models with polynomial growth in the diffusion term. We introduce a double implicit Milstein scheme and show that it possesses desirable properties. It converges strongly and preserves non-negativity for a rich family of financial models and can reproduce linear and nonlinear stability behaviour of the underlying SDE without severe restriction on the time step. Although the scheme is implicit, we point out examples of financial models where an explicit formula for the solution to the scheme can be found.

math.NA↗

arXiv2011

Lyapunov Exponents of Hybrid Stochastic Heat Equations

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 Markovian switching.

math.PR↗

arXiv2011

Competitive Lotka-Volterra Population Dynamics with Jumps

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 Lyapunov exponents; (c) Using a variation-of-constants formula for a class of SDEs with jumps, we provide explicit solution for 1-dimensional competitive Lotka-Volterra population dynamics with jumps, and investigate the sample Lyapunov exponent for each component and the extinction of our $n$-dimensional model.

math.PR↗

arXiv2010

Khasminskii-Type Theorem and LaSalle-Type Theorem for Stochastic Evolution Delay Equations

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 examples are established to illustrate the power of our theories.

math.PR↗

arXiv2009

Rate of convergence for numerical solutions to SFDEs with jumps

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 from the case of SFDEs without jumps where the order is 1/2 for any $p\ge 2$. It is therefore best to use the mean-square convergence for SFDEs with jumps. Consequently, under the local Lipschitz condition, we reveal that the order of the mean-square convergence is close to 1/2, provided that the local Lipschitz constants, valid on balls of radius $j$, do not grow faster than $\log j$.

math.PR↗