Research Interests:
Research Interests:
Research Interests:
Research Interests:
For Ait-Sahalia-type interest rate model with Poisson jumps, we are interested in strong convergence of a novel time-stepping method, called transformed jump-adapted backward Euler method (TJABEM). Under certain hypothesis, the considered... more
For Ait-Sahalia-type interest rate model with Poisson jumps, we are interested in strong convergence of a novel time-stepping method, called transformed jump-adapted backward Euler method (TJABEM). Under certain hypothesis, the considered model takes values in positive domain (0,∞). It is shown that the TJABEM can preserve the domain of the underlying problem. Furthermore, for the above model with non-globally Lipschitz drift and diffusion coefficients, the strong convergence rate of order one of the TJABEM is recovered with respect to a Lp-error criterion. Finally, numerical experiments are given to illustrate the theoretical results. Mathematics Subject Classification: 60H35, 60H15, 65C30.
Research Interests:
The aim of this study is the weak convergence rate of a temporal and spatial discretization scheme for stochastic Cahn–Hilliard equation with additive noise, where the spectral Galerkin method is used in space and the backward Euler... more
The aim of this study is the weak convergence rate of a temporal and spatial discretization scheme for stochastic Cahn–Hilliard equation with additive noise, where the spectral Galerkin method is used in space and the backward Euler scheme is used in time. The presence of the unbounded operator in front of the nonlinear term and the lack of the associated Kolmogorov equations make the error analysis much more challenging and demanding. To overcome these difficulties, we further exploit a novel approach proposed in [7] and combine it with Malliavin calculus to obtain an improved weak rate of convergence, in comparison with the corresponding strong convergence rates. The techniques used here are quite general and hence have the potential to be applied to other non-Markovian equations. As a byproduct the rate of the strong error can also be easily obtained.
Research Interests:
Research Interests:
A fractional trapezoidal rule type difference scheme for fractional order integro-differential equation is considered. The equation is discretized in time by means of a method based on the trapezoidal rule: while the time derivative is... more
A fractional trapezoidal rule type difference scheme for fractional order integro-differential equation is considered. The equation is discretized in time by means of a method based on the trapezoidal rule: while the time derivative is approximated by the standard trapezoidal rule, the integral term is discretized by means of a fractional quadrature rule constructed again from the trapezoidal rule. The solvability, stability and L2-norm convergence are proved. The convergence order is second order both in temporal and spatial directions. Furthermore, a spatial compact scheme, based on the fractional trapezoidal rule type difference scheme, is also proposed and the similar results are derived. The convergence order is second for time and fourth for space. Preliminary numerical experiment confirms our theoretical results.
Research Interests:
This paper is concerned with the analytic and numerical dissipativity of nonlinear integro-differential equations (IDEs). A dissipativity criteria for IDEs is given. It is shown that any A-stable linear 0-method for the systems is... more
This paper is concerned with the analytic and numerical dissipativity of nonlinear integro-differential equations (IDEs). A dissipativity criteria for IDEs is given. It is shown that any A-stable linear 0-method for the systems is dissipative. Numerical examples are given to confirm the theoretical results. © 2006 Elsevier Ltd. All rights reserved.
Research Interests:
Research Interests:
Research Interests:
Research Interests:
It is shown that spurious solutions are obtained when Runge-Kutta method is directly applied to the delay logistic equations with piecewise continuous arguments.Runge-Kutta method which does not admit spurious solutions is constructed.The... more
It is shown that spurious solutions are obtained when Runge-Kutta method is directly applied to the delay logistic equations with piecewise continuous arguments.Runge-Kutta method which does not admit spurious solutions is constructed.The convergent order of this method is investigated.It is shown that this method is locally asymptotically stable and globally asymptotically stable under certain conditions.
Research Interests:
Research Interests:
ABSTRACT In this paper, we analyze the weak error of a semi-discretization in time by the linear implicit Euler method for semilinear stochastic partial differential equations (SPDEs) with additive noise. The main result reveals how the... more
ABSTRACT In this paper, we analyze the weak error of a semi-discretization in time by the linear implicit Euler method for semilinear stochastic partial differential equations (SPDEs) with additive noise. The main result reveals how the weak order depends on the regularity of noise and that the order of weak convergence is twice that of strong convergence. In particular, the linear implicit Euler method for SPDEs driven by trace class noise achieves an almost optimal order 1−ϵ1−ϵ for arbitrarily small ϵ>0ϵ>0.
Research Interests:
Research Interests: Mathematics, Applied Mathematics, Numerical Analysis, Delay Differential Equation, Applied Mathematics and Computational Science, and 7 moreNumerical Analysis and Computational Mathematics, Linear Approximation, Numerical Stability, Differential equation, Integral Equation, Numerical Solution, and Electrical And Electronic Engineering
Research Interests:
Research Interests:
Research Interests:
Research Interests:
Research Interests:
Research Interests:
Research Interests:
Research Interests:
Research Interests:
Research Interests:
This paper is concerned with the stability of analytical and numerical solutions fornonlinearstochastic delay differential equations (SDDEs) with jumps. A sufficient condition for mean-square exponential stability of the exact solution is... more
This paper is concerned with the stability of analytical and numerical solutions fornonlinearstochastic delay differential equations (SDDEs) with jumps. A sufficient condition for mean-square exponential stability of the exact solution is derived. Then, mean-square stability of the numerical solution is investigated. It is shown that the compensated stochastic θ methods inherit stability property of the exact solution. More precisely, the methods are mean-square stable for any stepsizeΔt=τ/mwhen1/2≤θ≤1, and they are exponentially mean-square stable if the stepsizeΔt∈(0,Δt0)when0≤θ<1. Finally, some numerical experiments are given to illustrate the theoretical results.
Research Interests:
This paper is concerned with mean square exponential stability of switched stochastic system with interval time-varying delays. The time delay is any continuous function belonging to a given interval, but not necessary to be... more
This paper is concerned with mean square exponential stability of switched stochastic system with interval time-varying delays. The time delay is any continuous function belonging to a given interval, but not necessary to be differentiable. By constructing a suitable augmented Lyapunov-Krasovskii functional combined with Leibniz-Newton’s formula, a switching rule for the mean square exponential stability of switched stochastic system with interval time-varying delays and new delay-dependent sufficient conditions for the mean square exponential stability of the switched stochastic system are first established in terms of LMIs. Numerical example is given to show the effectiveness of the obtained result.
Research Interests:
Abstract. This paper is devoted to the convergence analysis of stochastic θ-methods for non-linear neutral stochastic differential delay equations (NSDDEs) in Itô sense. The basic idea is to reformulate the original problem eliminating... more
Abstract. This paper is devoted to the convergence analysis of stochastic θ-methods for non-linear neutral stochastic differential delay equations (NSDDEs) in Itô sense. The basic idea is to reformulate the original problem eliminating the dependence on the differentiation of ...
Research Interests:
Research Interests:
This article aims to establish a fundamental mean-square convergence theorem for general one-step numerical approximations of Lévy noise driven stochastic differential equations (SDEs), with non-globally Lipschitz coefficients. As... more
This article aims to establish a fundamental mean-square convergence theorem for general one-step numerical approximations of Lévy noise driven stochastic differential equations (SDEs), with non-globally Lipschitz coefficients. As applications of the obtained fundamental convergence theorem, two novel explicit schemes are designed and their convergence rates are exactly identified for problems subject to both multiplicative and additive noises. Distinct from existing works, we do not impose a globally Lipschitz condition on the jump coefficient but formulate appropriate assumptions to allow for its super-linear growth. A price to pay for such a more relaxed condition is assuming finite Lévy measure. Moreover, new arguments are developed to handle essential difficulties in the convergence analysis, caused by the super-linear growth of the jump coefficient and the fact that higher moment bounds of the Poisson increments ∫_t^t+h∫_Z N̅(ds,dz), t ≥ 0, h >0 contribute to magnitude not ...
Research Interests:
Novel fully discrete schemes are developed to numerically approximate a semilinear stochastic wave equation driven by additive space-time white noise. Spectral Galerkin method is proposed for the spatial discretization, and exponential... more
Novel fully discrete schemes are developed to numerically approximate a semilinear stochastic wave equation driven by additive space-time white noise. Spectral Galerkin method is proposed for the spatial discretization, and exponential time integrators involving linear functionals of the noise are introduced for the temporal approximation. The resulting fully discrete schemes are very easy to implement and allow for higher strong convergence rate in time than existing time-stepping schemes such as the Crank-Nicolson-Maruyama scheme and the stochastic trigonometric method. Particularly, it is shown that the new schemes achieve in time an order of 1- ϵ for arbitrarily small ϵ >0, which exceeds the barrier order 1/2 established by Walsh. Numerical results confirm higher convergence rates and computational efficiency of the new schemes.
Research Interests:
Research Interests:
Symmetric method and symplectic method are classical notions in the theory of Runge-Kutta methods. They can generate numerical flows that respectively preserve the symmetry and symplecticity of the continuous flows in the phase space.... more
Symmetric method and symplectic method are classical notions in the theory of Runge-Kutta methods. They can generate numerical flows that respectively preserve the symmetry and symplecticity of the continuous flows in the phase space. Adjoint method is an important way of constructing a new Runge-Kutta method via the symmetrisation of another Runge-Kutta method. In this paper, we introduce a new notion, called symplectic-adjoint Runge-Kutta method. We prove some interesting properties of the symmetric-adjoint and symplectic-adjoint methods. These properties reveal some intrinsic connections among several classical classes of Runge-Kutta methods. In particular, the newly introduced notion and the corresponding properties enable us to develop a novel and practical approach of constructing high-order explicit Runge-Kutta methods, which is a challenging and longly overlooked topic in the theory of Runge-Kutta methods.
Research Interests:
We discretize the stochastic Allen-Cahn equation with additive noise by means of a spectral Galerkin method in space and a tamed version of the exponential Euler method in time. The resulting error bounds are analyzed for the... more
We discretize the stochastic Allen-Cahn equation with additive noise by means of a spectral Galerkin method in space and a tamed version of the exponential Euler method in time. The resulting error bounds are analyzed for the spatio-temporal full discretization in both strong and weak senses. Different from existing works, we develop a new and direct approach for the weak error analysis, which does not rely on the associated Kolmogorov equation. It turns out that the obtained weak convergence rates are, in both spatial and temporal direction, essentially twice as high as the strong convergence rates. Also, it is revealed how the weak convergence rates depend on the regularity of the noise. Numerical experiments are finally reported to confirm the theoretical conclusion.
Research Interests:
ABSTRACT
Research Interests:
Abstract Traditional explicit schemes such as the Euler-Maruyama, Milstein and stochastic Runge-Kutta methods, in general, result in strong and weak divergence when solving stochastic differential equations (SDEs) with super-linearly... more
Abstract Traditional explicit schemes such as the Euler-Maruyama, Milstein and stochastic Runge-Kutta methods, in general, result in strong and weak divergence when solving stochastic differential equations (SDEs) with super-linearly growing coefficients. Motivated by this, various modified versions of explicit Euler and Milstein methods were constructed and analyzed in the literature. In the present paper, we aim to introduce a family of explicit tamed stochastic Runge-Kutta (TSRK) methods for commutative SDEs with super-linearly growing drift and diffusion coefficients. Strong convergence rates of order 1.0 are successfully identified for the proposed methods under certain non-globally Lipschitz conditions. Compared to the Milstein-type methods involved with derivatives of coefficients, the newly proposed derivative-free TSRK methods can be computationally more efficient. Numerical experiments are reported to confirm the expected strong convergence rate of the TSRK methods.
Research Interests:
Research Interests:
Research Interests:
Research Interests:
Research Interests:
Research Interests:
... 26] further studied the nonlinear stability of two classes of adapted RungeKutta methods for the same class of stiff VDIDEs. ... For the delay differential equations (DDEs) with constant delay, Huang [9] gave a sufficient condition... more
... 26] further studied the nonlinear stability of two classes of adapted RungeKutta methods for the same class of stiff VDIDEs. ... For the delay differential equations (DDEs) with constant delay, Huang [9] gave a sufficient condition for the dissipativity of theoretical solution, and ...
Research Interests:
Research Interests:
This paper is concerned with the numerical solution of functional-diere n tial and functional equations which include functional-diere n tial equations of neutral type as special cases. The adaptation of general linear methods is... more
This paper is concerned with the numerical solution of functional-diere n tial and functional equations which include functional-diere n tial equations of neutral type as special cases. The adaptation of general linear methods is considered. It is proved that A-stable general linear methods can inherit the asymptotic stability of underlying linear systems. Some general results of numerical stability are also given.
ABSTRACT
Research Interests:
ABSTRACT