Split-step Backward Euler Method Research Articles

Numerical Analysis of Split-Step Backward Euler Method with Truncated Wiener Process for a Stochastic Susceptible-Infected-Susceptible Model.

This article deals with the numerical positivity, boundedness, convergence, and dynamical behaviors for stochastic susceptible-infected-susceptible (SIS) model. To guarantee the biological significance of the split-step backward Euler method applied to the stochastic SIS model, the numerical positivity and boundedness are investigated by the truncated Wiener process. Motivated by the almost sure boundedness of exact and numerical solutions, the convergence is discussed by the fundamental convergence theorem with a local Lipschitz condition. Moreover, the numerical extinction and persistence are initially obtained by an exponential presentation of the stochastic stability function and strong law of the large number for martingales, which reproduces the existing theoretical results. Finally, numerical examples are given to validate our numerical results for the stochastic SIS model.

Finite-volume approximation of the invariant measure of a viscous stochastic scalar conservation law

Abstract We study the numerical approximation of the invariant measure of a viscous scalar conservation law, one-dimensional and periodic in the space variable and stochastically forced with a white-in-time but spatially correlated noise. The flux function is assumed to be locally Lipschitz continuous and to have at most polynomial growth. The numerical scheme we employ discretizes the stochastic partial differential equation (SPDE) according to a finite-volume method in space and a split-step backward Euler method in time. As a first result, we prove the well posedness as well as the existence and uniqueness of an invariant measure for both the semidiscrete and the split-step scheme. Our main result is then the convergence of the invariant measures of the discrete approximations, as the space and time steps go to zero, towards the invariant measure of the SPDE, with respect to the second-order Wasserstein distance. We investigate rates of convergence theoretically, in the case where the flux function is globally Lipschitz continuous with a small Lipschitz constant, and numerically for the Burgers equation.

Strong convergence of the split-step backward Euler method for stochastic delay differential equations with a nonlinear diffusion coefficient

This paper is concerned with the strong convergence of numerical methods for stochastic differential equations with a variable delay. Based on the concepts of stochastic C-stability and B-consistency, we first derive a fundamental strong convergence theorem for general one-step methods applied to nonlinear autonomous stochastic delay differential equations under a local Lipschitz condition and a coupled condition on the drift and diffusion coefficients. These conditions admit that the diffusion coefficient is highly nonlinear. Then, we construct the split-step backward Euler method. It is verified that the method is stochastically C-stable and B-consistent of order 1/2 so that the method is strongly convergent of order 1/2 for the considered problems. Finally, the theoretical results are confirmed by numerical experiments.

Mean-square dissipative methods for stochastic age-dependent capital system with fractional Brownian motion and jumps

In this paper, we analyze mean-square dissipativity of numerical methods applied to a class of stochastic age-dependent (vintage) capital system with fractional Brownian motion (fBm) and Poisson jumps. Some sufficient conditions are obtained for ensuring the underlying systems are mean-square dissipative. It is shown that the mean-square dissipativity is preserved by the compensated split-step backward Euler method and compensated backward Euler method without any restriction on stepsize, while the split-step backward Euler method and backward Euler method could reproduce mean-square dissipativity under a stepsize constraint. Those results indicate that compensated numerical methods achieve superiority over non-compensated numerical methods in terms of mean-square dissipativity. A numerical example is provided to illustrate the theoretical results.

Analysis of stability for stochastic delay integro-differential equations

In this paper, we concern stability of numerical methods applied to stochastic delay integro-differential equations. For linear stochastic delay integro-differential equations, it is shown that the mean-square stability is derived by the split-step backward Euler method without any restriction on step-size, while the Euler–Maruyama method could reproduce the mean-square stability under a step-size constraint. We also confirm the mean-square stability of the split-step backward Euler method for nonlinear stochastic delay integro-differential equations. The numerical experiments further verify the theoretical results.

Asymptotic mean-square boundedness of the numerical solutions of stochastic age-dependent population equations with Poisson jumps

This paper focuses on asymptotic mean-square boundedness of several numerical methods applied to a class of stochastic age-dependent population equations with Poisson jumps. The conditions under which the underlying systems are asymptotic mean-square boundedness are considered. It is shown that the asymptotic mean-square boundedness is preserved by the compensated split-step backward Euler method and compensated backward Euler method without any restriction on stepsize, while the split-step backward Euler method and backward Euler method could reproduce asymptotic mean-square boundedness under a stepsize constraint. The results indicate that compensated numerical methods achieve superiority over non-compensated numerical methods in terms of asymptotic mean-square boundedness. Finally, an example is given for illustration.

Exponential stability of the split-step θ-method for neutral stochastic delay differential equations with jumps

The exponential mean-square stability of the split-step θ-method for neutral stochastic delay differential equations (NSDDEs) with jumps is considered. New conditions for jumps are proposed to ensure the exponential mean-square stability of the trivial solution. If the drift coefficient satisfies the linear growth condition, it is shown that the split-step θ-method can reproduce the exponential mean-square stability of the trivial solution for the constrained stepsize. Then by applying the Chebyshev inequality and the Borel–Cantelli lemma, the almost sure exponential stability of both the trivial solution and the numerical solution can be obtained. Since split-step θ-method covers Euler–Maruyama (EM) method and split-step backward Euler (SSBE) method, the conclusions are valid for these two methods. Moreover, they can adapt to the NSDDEs and the SDDEs with jumps. Finally, a numerical example illustrates the effectiveness of the theoretical results.

Mean-square dissipativity of numerical methods for a class of resource-competition models with fractional Brownian motion

ABSTRACTThe concept of dissipativity in dynamical systems generalizes the idea of a Lyapunov stability. In this paper the dissipativity is used to designate that the system possesses a bounded absorbing set. Specifically, this paper studies mean-square dissipativity of two numerical methods for a class of resource-competition models with fractional Brownian motion (fBm). Some conditions under which the underlying systems are mean-square dissipative are established. It is shown that the mean-square dissipativity is preserved by the split-step θ-method (SSθ) under some constraints, while the split-step backward Euler method (SSBE) could inherit mean-square dissipativity without any restriction on stepsize. The results of this study indicate that the split-step backward Euler method (SSBE) outperforms the split-step θ-method (SSθ) in terms of mean-square dissipativity. Finally, an example is given to illustrate the effectiveness of the results.

Stochastic C-Stability and B-Consistency of Explicit and Implicit Milstein-Type Schemes

This paper focuses on two variants of the Milstein scheme, namely the split-step backward Milstein method and a newly proposed projected Milstein scheme, applied to stochastic differential equations which satisfy a global monotonicity condition. In particular, our assumptions include equations with super-linearly growing drift and diffusion coefficient functions and we show that both schemes are mean-square convergent of order 1. Our analysis of the error of convergence with respect to the mean-square norm relies on the notion of stochastic C-stability and B-consistency, which was set up and applied to Euler-type schemes in [Beyn, Isaak, Kruse, J. Sci. Comp., 2015]. As a direct consequence we also obtain strong order 1 convergence results for the split-step backward Euler method and the projected Euler-Maruyama scheme in the case of stochastic differential equations with additive noise. Our theoretical results are illustrated in a series of numerical experiments.

Stochastic C-Stability and B-Consistency of Explicit and Implicit Euler-Type Schemes

This paper is concerned with the numerical approximation of stochastic ordinary differential equations, which satisfy a global monotonicity condition. This condition includes several equations with super-linearly growing drift and diffusion coefficient functions such as the stochastic Ginzburg-Landau equation and the 3/2-volatility model from mathematical finance. Our analysis of the mean-square error of convergence is based on a suitable generalization of the notions of C-stability and B-consistency known from deterministic numerical analysis for stiff ordinary differential equations. An important feature of our stability concept is that it does not rely on the availability of higher moment bounds of the numerical one-step scheme. While the convergence theorem is derived in a somewhat more abstract framework, this paper also contains two more concrete examples of stochastically C-stable numerical one-step schemes: the split-step backward Euler method from Higham et al. (2002) and a newly proposed explicit variant of the Euler-Maruyama scheme, the so called projected Euler-Maruyama method. For both methods the optimal rate of strong convergence is proven theoretically and verified in a series of numerical experiments.

Mean-square dissipativity of numerical methods for a class of stochastic neural networks with fractional Brownian motion and jumps

In this paper, we introduce a class of stochastic neural networks with fractional Brownian motion (fBm) and Poisson jumps. We also concern mean-square dissipativity of numerical methods applied to a class of stochastic neural networks with fBm and jumps. The conditions under which the underlying systems are mean-square dissipative are considered. It is shown that the mean-square dissipativity is preserved by the compensated split-step backward Euler method and compensated backward Euler method without any restriction on stepsize, while the split-step backward Euler method and backward Euler method could reproduce mean-square dissipativity under a stepsize constraint. The results indicate that compensated numerical methods achieve superiority over non-compensated numerical methods in terms of mean-square dissipativity. Finally, an example is given for illustration.

Implicit numerical methods for highly nonlinear neutral stochastic differential equations with time-dependent delay

This paper represents the continuation of the analysis from papers Milošević (2011) [10] and Milošević (2013) [11]. The main aim of this paper is to establish certain results for the backward Euler method for a class of neutral stochastic differential equations with time-dependent delay. For that purpose, the split-step backward Euler method, which represents an extension of the backward Euler method, is introduced for this class of equations. Conditions under which the split-step backward Euler method, and thus the backward Euler method, is well defined are revealed. Moreover, the convergence in probability of the backward Euler method is proved under certain nonlinear growth conditions including the one-sided Lipschitz condition. This result is proved using the technique which is based on the application of the continuous-time approximation. For this reason, the discrete forward–backward Euler method is involved since it allows its continuous version to be well defined from the aspect of measurability. The convergence in probability is established for the continuous forward–backward Euler solution, which is essential for proving the same result for both discrete forward–backward and backward Euler methods. Additionally, it is proved that the discrete backward Euler equilibrium solution is globally a.s. asymptotically exponentially stable, without the linear growth condition on the drift coefficient of the equation. As usual, the whole consideration is affected by the presence and properties of the delay function.

Mean-square dissipativity of several numerical methods for stochastic differential equations with jumps

This paper focuses on mean-square dissipativity of several numerical methods applied to a class of stochastic differential equations with jumps. The conditions under which the underlying systems are mean-square dissipative are given. It is shown that the mean-square dissipativity is preserved by the compensated split-step backward Euler method and compensated backward Euler method without any restriction on stepsize, while the split-step backward Euler method and backward Euler method could reproduce mean-square dissipativity under a stepsize constraint. Those results indicate that compensated numerical methods achieve superiority over non-compensated numerical methods in terms of mean-square dissipativity.

Numerical methods for nonlinear stochastic delay differential equations with jumps

In this paper, we modified the split-step backward Euler method (MSSBE) for stochastic delay differential equations with Poisson-driven jumps (SDDEwJs). Second, we prove that MSSBE is strongly convergent if the drift coefficient f(x,y) satisfies one-side Lipschitz with respect to x, global Lipschitz with respect to y, the diffusion and jump coefficients are globally Lipschitz. On the way to proving the convergence result, we show that Euler–Maruyama method converges strongly when SDDEwJs coefficients satisfy local Lipschitz condition, the pth moments of the exact and numerical solution are bounded for some p>2; the MSSBE may be viewed as an Euler–Maruyama approximation to a perturbed SDDEwJs of the same form.

Almost Sure andLpConvergence of Split-Step Backward Euler Method for Stochastic Delay Differential Equation

The convergence of the split-step backward Euler (SSBE) method applied to stochastic differential equation with variable delay is proven inLp-sense. Almost sure convergence is derived from theLpconvergence by Chebyshev’s inequality and the Borel-Cantelli lemma; meanwhile, the convergence rate is obtained.

The Semimartingale Approach to Almost Sure Stability Analysis of a Two-Stage Numerical Method for Stochastic Delay Differential Equation

Almost sure exponential stability of the split-step backward Euler (SSBE) method applied to an Itô-type stochastic differential equation with time-varying delay is discussed by the techniques based on Doob-Mayer decomposition and semimartingale convergence theorem. Numerical experiments confirm the theoretical analysis.

A note on stability of the split-step backward Euler method for linear stochastic delay integro-differential equations

In the literature (Tan and Wang, 2010), Tan and Wang investigated the convergence of the split-step backward Euler (SSBE) method for linear stochastic delay integro-differential equations (SDIDEs) and proved the mean-square stability of SSBE method under some condition. Unfortunately, the main result of stability derived by the condition is somewhat restrictive to be applied for practical application. This paper improves the corresponding results. The authors not only prove the mean-square stability of the numerical method but also prove the general mean-square stability of the numerical method. Furthermore, an example is given to illustrate the theory.

Stability in the numerical simulation of stochastic delayed Hopfield neural networks

This paper is concerned with the mean-square stability of the Split-Step Backward Euler method for stochastic delayed Hopfield neural networks. The sufficient conditions to guarantee the mean-square stability of the Split-Step Backward Euler method are given. Moreover, an example of the comparison of our method with the Euler–Maruyama method is used to show the superiority of our method.

Strong convergence of split-step backward Euler method for stochastic differential equations with non-smooth drift

In this paper, we are concerned with the numerical approximation of stochastic differential equations with discontinuous/nondifferentiable drifts. We show that under one-sided Lipschitz and general growth conditions on the drift and global Lipschitz condition on the diffusion, a variant of the implicit Euler method known as the split-step backward Euler (SSBE) method converges with strong order of one half to the true solution. Our analysis relies on the framework developed in [D. J. Higham, X. Mao and A. M. Stuart, Strong convergence of Euler-type methods for nonlinear stochastic differential equations, SIAM Journal on Numerical Analysis, 40 (2002) 1041–1063] and exploits the relationship which exists between explicit and implicit Euler methods to establish the convergence rate results.

The improved split-step backward Euler method for stochastic differential delay equations

A new, improved split-step backward Euler method is introduced and analysed for stochastic differential delay equations (SDDEs) with generic variable delay. The method is proved to be convergent in the mean-square sense under conditions (Assumption 3.1) that the diffusion coefficient g(x, y) is globally Lipschitz in both x and y, but the drift coefficient f(x, y) satisfies the one-sided Lipschitz condition in x and globally Lipschitz in y. Further, the exponential mean-square stability of the proposed method is investigated for SDDEs that have a negative one-sided Lipschitz constant. Our results show that the method has the unconditional stability property, in the sense, that it can well reproduce stability of the underlying system, without any restrictions on stepsize h. Numerical experiments and comparisons with existing methods for SDDEs illustrate the computational efficiency of our method.