Backward Euler-Maruyama Method Research Articles

Stability in the small moment sense of the backward Euler–Maruyama method for stochastic differential equations with super-linear coefficients

For stochastic differential equations (SDEs) with super-linear drift and diffusion coefficients, the backward Euler–Maruyama (BEM) method is considered to reproduce the stability of the underlying SDEs. The pth moment exponential stability for some small p∈(0,1) and the almost sure exponential stability of the BEM method are proved. The results in this paper partially extend those in Higham, Mao and Yuan (2007). Numerical simulations are provided to illustrate the theoretical results.

Ergodicity and approximations of invariant measures for stochastic lattice systems with Markovian switching

This paper is concerned with the dynamics of stochastic lattice systems with Markovian switching. Based on the well-posedness of solutions, we first prove the ergodicity of invariant measures and show that the Markov chain facilitates the existence of invariant measures. In order to investigate numerical invariant measures, the convergence of invariant measures is considered between the original systems and their finite-dimensional truncated systems. Due to this, we can use the backward Euler-Maruyama method to approximate invariant measures for such infinite-dimensional systems. This work provides a feasible path for the convergence of the finite-dimensional numerical invariant measures to the analytical invariant measure.

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.

General decay stability of backward Euler–Maruyama method for nonlinear stochastic integro-differential equations

The general decay stability of backward Euler–Maruyama(EM) method is discussed for stochastic integro-differential equations(SIDEs) in this paper. Under polynomial growth condition, both continuous and discrete solutions admit high nonlinearity of the underlying equations. Besides, the considered stability namely the general decay stability is a generalized stability including almost sure polynomially stability and almost sure exponentially stability. Making using of the nonnegative semi-martingale convergence theorem, for SIDEs we propose sufficient conditions to obtain the general decay stability of backward EM method.

Backward Euler–Maruyama Method for the Random Periodic Solution of a Stochastic Differential Equation with a Monotone Drift

In this paper, we study the existence and uniqueness of the random periodic solution for a stochastic differential equation with a one-sided Lipschitz condition (also known as monotonicity condition) and the convergence of its numerical approximation via the backward Euler–Maruyama method. The existence of the random periodic solution is shown as the limit of the pull-back flows of the SDE and the discretized SDE, respectively. We establish a convergence rate of the strong error for the backward Euler–Maruyama method with order of convergence 1/2.

Error estimates of the backward Euler–Maruyama method for multi-valued stochastic differential equations

In this paper we derive error estimates of the backward Euler–Maruyama method applied to multi-valued stochastic differential equations. An important example of such an equation is a stochastic gradient flow whose associated potential is not continuously differentiable but assumed to be convex. We show that the backward Euler–Maruyama method is well-defined and convergent of order at least 1/4 with respect to the root-mean-square norm. Our error analysis relies on techniques for deterministic problems developed in Nochetto et al. (Commun Pure Appl Math 53(5):525–589, 2000). We verify that our setting applies to an overdamped Langevin equation with a discontinuous gradient and to a spatially semi-discrete approximation of the stochastic p-Laplace equation.

Stability of numerical solutions for the stochastic pantograph differential equations with variable step size

In this paper, two different techniques are used to research the stability of numerical solutions with variable step size for stochastic pantograph differential equations. Based on the discrete semimartingale convergence theorem, the almost sure polynomial stability and the αth moment polynomial stability of the numerical solutions by the Euler–Maruyama (EM) and the backward Euler–Maruyama (BEM) methods with variable step size have been discussed. By using the discrete Razumikhin-type technique, the global αth moment asymptotically stability and the αth moment polynomial stability of the numerical solutions by the general numerical scheme with variable step size have been researched. Based on the results of the stability of the general numerical scheme, we discuss the stability of two special numerical solutions, namely the EM method and the BEM method. By comparing the two techniques (i.e. the discrete semimartingale convergence theorem and the discrete Razumikhin-type technique) on the research of the moment stability, we can easily get the latter is better. To illustrate the theoretical results, we give some examples to examine the almost sure polynomial stability and the αth moment polynomial stability of the numerical solutions by the EM and BEM schemes with variable step size.

Razumikhin-type theorems on the moment stability of the exact and numerical solutions for the stochastic pantograph differential equations

In this paper, we establish two different α th moment ψ -type stability criterions on the exact and numerical solutions of the stochastic pantograph differential equations. One is established by the virtue of Lyapunov method, and the other is used by the continuous and discrete Razumikhin technique. By comparing the two stability criterions, we can easily see that the conditions constructed by the Razumikhin-type technique are better than those constructed by the virtue of Lyapunov method. Using the conditions constructed for the α th moment ψ -type stability, we study the stability of the Euler–Maruyama method and the backward Euler–Maruyama method, respectively, for a special class of the stochastic pantograph differential equations. Finally, examples are given to illustrate the consistence with the theoretical results on the α th moment ψ -type stability.

Razumikhin-type technique on stability of exact and numerical solutions for the nonlinear stochastic pantograph differential equations

In this paper, we establish Razumikhin-type theorems on $$\alpha $$ th moment polynomial stability of exact solution for the stochastic pantograph differential equations, which improves the existing stochastic Razumikhin-type theorems. By using discrete Razumikhin-type technique, we construct conditions for the stability of general numerical scheme of the stochastic pantograph differential equations (SPDEs). The stabilities mainly conclude the global $$\alpha $$ th moment asymptotically stability and $$\alpha $$ th moment polynomial stability. Using the conditions constructed for the stability of the numerical solutions, we discuss the stability of two special numerical methods, namely the Euler–Maruyama method and the backward Euler–Maruyama method. Finally, an example is given to illustrate the consistence with the theoretical results on $$\alpha $$ th moment polynomial stability.

Backward Euler-Maruyama method applied to nonlinear hybrid stochastic differential equations with time-variable delay

In this paper, we consider strong convergence andalmost sure exponential stability of the backward Euler-Maruyamamethod for nonlinear hybrid stochastic differential equationswith time-variable delay. Under the local Lipschitz condition andpolynomial growth condition, it is proved that the backwardEuler-Maruyama method is strongly convergent. Additionally, themoment estimates and almost sure exponential stability for theanalytical solution are proved. Also, under the appropriatecondition, we show that the numerical solutions for the backwardEuler-Maruyama methods are almost surely exponentially stable. A numerical experiment is given to illustrate thecomputational effectiveness and the theoretical results of themethod.

Almost sure exponential stability of numerical solutions for stochastic pantograph differential equations

The sufficient conditions of the almost sure exponential stability of the exact solution for the stochastic pantograph differential equation are considered, with a Khasminskii-type condition. The almost sure exponential stability of the numerical solutions by the Euler–Maruyama method and the backward Euler–Maruyama method is also discussed, based on the discrete semimartingale convergence theorem. We present the sufficient conditions for the stability of the Euler–Maruyama method, with one extra condition when compared with the exact solution. We show that the backward Euler–Maruyama method can share almost the same conditions for the almost sure exponential stability as the exact solution.

Strong convergence of implicit numerical methods for nonlinear stochastic functional differential equations

Abstract The main aim of this work is to prove that the backward Euler–Maruyama approximate solutions converge strongly to the true solutions for stochastic functional differential equations with superlinear growth coefficients. The paper also gives the boundedness and mean-square exponential stability of the exact solutions, and shows that the backward Euler–Maruyama method can preserve the boundedness of mean-square moments. Finally, a highly nonlinear example is provided to illustrate the main results.

Mean‐square stability of the backward Euler–Maruyama method for neutral stochastic delay differential equations with jumps

This paper is mainly considered whether the mean‐square stability of neutral stochastic delay differential equations (NSDDEs) with jumps is shared with that of the backward Euler–Maruyama method. Under the one‐sided Lipschitz condition and the linear growth condition, the trivial solution of NSDDEs with jumps is proved to be mean‐square stable by using the functional comparison principle and the Barbalat's lemma. It is shown that the backward Euler–Maruyama method can reproduce the mean‐square stability of the trivial solution under the same conditions. The implicit backward Euler–Maruyama method shows better characteristic than the explicit Euler–Maruyama method for the reason that it works without the linear growth condition on the drift coefficient. Compared with some existing results, our results do not need to add extra condition on the neutral part. The conclusions can be applied to NSDDEs and SDDEs with jumps. The effectiveness of the theoretical results is illustrated by an example. Copyright © 2016 John Wiley & Sons, Ltd.

The moment exponential stability criterion of nonlinear hybrid stochastic differential equations and its discrete approximations

Based on the martingale theory and large deviation techniques, we investigate the pth moment exponential stability criterion of the exact and numerical solutions to hybrid stochastic differential equations (SDEs) under the local Lipschitz condition. This new stability criterion shows that Markovian switching can serve as a stochastic stabilizing factor by its logarithmic moment-generating function. We also investigate the pth moment exponential stability of Euler–Maruyama (EM), backward EM (BEM) and split-step backward EM (SSBEM) approximations for hybrid SDEs and show that, under the additional linear growth condition, the EM method can share the mean-square exponential stability of the exact solution for sufficiently small step size. However, the BEM method can work without the linear growth condition. We further investigate the SSBEM method under a coupled condition.

A discrete stochastic Gronwall lemma

The purpose of this paper is the derivation of a discrete version of the stochastic Gronwall lemma involving a martingale. The proof is based on a corresponding deterministic version of the discrete Gronwall lemma and an inequality bounding the supremum in terms of the infimum for discrete time martingales. As an application the proof of an a priori estimate for the backward Euler–Maruyama method is included.

Mean-square convergence of the BDF2-Maruyama and backward Euler schemes for SDE satisfying a global monotonicity condition

In this paper the numerical approximation of stochastic differential equations satisfying a global monotonicity condition is studied. The strong rate of convergence with respect to the mean square norm is determined to be $\frac{1}{2}$ for the two-step BDF-Maruyama scheme and for the backward Euler-Maruyama method. In particular, this is the first paper which proves a strong convergence rate for a multi-step method applied to equations with possibly superlinearly growing drift and diffusion coefficient functions. We also present numerical experiments for the $\tfrac32$-volatility model from finance, which verify our results in practice and indicate that the BDF2-Maruyama method offers advantages over Euler-type methods if the stochastic differential equation is stiff or driven by a noise with small intensity.

Numerical stationary distribution and its convergence for nonlinear stochastic differential equations

To avoid finding the stationary distributions of stochastic differential equations by solving the nontrivial Kolmogorov–Fokker–Planck equations, the numerical stationary distributions are used as the approximations instead. This paper is devoted to approximate the stationary distribution of the underlying equation by the Backward Euler–Maruyama method. Currently existing results (Mao et al., 2005; Yuan et al., 2005; Yuan et al., 2004) are extended in this paper to cover larger range of nonlinear SDEs when the linear growth condition on the drift coefficient is violated.

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.

Almost sure exponential stability of the backward Euler–Maruyama discretization for highly nonlinear stochastic functional differential equation

Stability of numerical solutions to stochastic differential equations have received an increasing attention, but there is so far no work on stability of numerical solutions to nonlinear stochastic functional differential equations (SFDEs). To close the gap, the paper develops a criterion on stability of numerical solutions to highly nonlinear SFDEs. By using of the discrete semi-martingale convergence theorem, we show that backward Euler–Maruyama method can reproduce almost sure exponential stability of the exact solutions to highly nonlinear SFDEs. Several high order examples are provided to illustrate the main results, which implies that a wide class of nonlinear systems obeys the new criterion.

Almost Surely Exponential Stability of Numerical Solutions for Stochastic Pantograph Equations

Our effort is to develop a criterion on almost surely exponential stability of numerical solution to stochastic pantograph differential equations, with the help of the discrete semimartingale convergence theorem and the technique used in stable analysis of the exact solution. We will prove that the Euler-Maruyama (EM) method can preserve almost surely exponential stability of stochastic pantograph differential equations under the linear growth conditions. And the backward EM method can reproduce almost surely exponential stability for highly nonlinear stochastic pantograph differential equations. A highly nonlinear example is provided to illustrate the main theory.