Convergence In Mean-square Sense Research Articles

Anti-periodic motion and mean-square exponential convergence of nonlocal discrete-time stochastic competitive lattice neural networks with fuzzy logic

This paper firstly establishes the discrete-time lattice networks for nonlocal stochastic competitive neural networks with reaction diffusions and fuzzy logic by employing a mix techniques of finite difference to space variables and Mittag-Leffler time Euler difference to time variable. The proposed networks consider both the effects of spatial diffusion and fuzzy logic, whereas most of the existing literatures focus only on discrete-time networks without spatial diffusion. Firstly, the existence of a unique ω-anti-periodic in distribution to the networks is addressed by employing Banach contractive mapping principle and the theory of stochastic calculus. Secondly, global exponential convergence in mean-square sense to the networks is discussed on the basis of constant variation formulas for sequences. Finally, an illustrative example is used to show the feasible of the works in the current paper with the help of MATLAB Toolbox. The work in this paper is pioneering in this regard and it has created a certain research foundations for future studies in this area.

Strong 1.5 order scheme for second-order stochastic differential equations without Levy area

In this work, a finite difference scheme combined with a truncated spectral expansion of white noise is presented for solving second-order stochastic differential equations (SDEs) with additive white noise. Consistency of the approximate equation, which is obtained by replacing the white noise with its spectral truncation, is considered in mean-square sense. Based on it, a full discrete scheme is constructed and the mean-square convergence order of numerical solution is investigated. Numerical examples show that the presented scheme is of 1.5-order convergence in mean-square sense, which verifies the theoretical findings and is a half order higher than converting the equation to a two-dimensional system with additive white noise and using the Euler-Maruyama scheme to solve the resulting problem. It is also illustrated that the piecewise version of the spectral approximation of white noise plus proper time discretization works well for long-time simulation.

On numerical methods to second-order singular initial value problems with additive white noise

In this work, we investigate the strong convergence of the Euler–Maruyama method for second-order stochastic singular initial value problems with additive white noise. The singularity at the origin brings a big challenge that the classical framework for stochastic differential equations and numerical schemes cannot work. By converting the problem to a first-order stochastic singular differential system, the existence and uniqueness of the exact solution is studied. Moreover, under some suitable assumptions, it is proved that the Euler–Maruyama method is of (1/2−ɛ) order convergence in mean-square sense, where ɛ is an arbitrarily small positive number, which is different from the consensus that the Euler–Maruyama method is convergent with first order in strong sense when solving stochastic differential equations with additive white noise. While, it is found that if the diffusion coefficient vanishes at the origin, the convergence order in mean-square sense will be increased to 1−ɛ. Our theoretical findings are well verified by numerical examples.

Strong convergence of explicit schemes for highly nonlinear stochastic differential equations with Markovian switching

Two projected Euler type schemes are analyzed for stochastic differential equations with Markovian switching whose coefficients are super-linear. Under the polynomial growth condition and the monotone condition, we investigate the convergence in mean square sense of these numerical methods. Besides, we also discuss the convergence rates of these two schemes for highly nonlinear equations (including stochastic differential equations with and without Markovian switching) with small noise. Finally, some numerical experiments are given to verify our theoretical results.

On strong convergence of explicit numerical methods for stochastic delay differential equations under non-global Lipschitz conditions

In this paper, we study the convergence of explicit numerical methods in strong sense for stochastic delay differential equations (SDDEs) with super-linear growth coefficients. Under non-globally Lipschitz conditions, a fundamental theorem on convergence has been constructed to elaborate the relationship of convergence rate between the local truncated error and the global error of one-step explicit methods in the sense of pth moments. A class of balanced Euler schemes has been presented and the boundedness of numerical solutions has been proved. By using the fundamental theorem, we prove that the balanced Euler scheme is of 0.5 order convergence in mean-square sense. Numerical examples verify the theoretical predictions.

Almost sure convergence of a semidiscrete Milstein scheme for SPDEs of Zakai type

A semidiscrete Milstein scheme for stochastic partial differential equations of Zakai type on a bounded domain of is derived. It is shown that the order of convergence of this scheme is 1 for convergence in mean square sense. For almost sure convergence, the order of convergence is proved to be for any .