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The present work aims to analyze mean-square convergence rates of split-step theta Milstein methods with method parameters θ ∈ [ 1 2 , 1 ] for stochastic differential equations with non-globally Lipschitz diffusion coefficients. The expected convergence rate of order one is successfully proved under more relaxed conditions, compared to existing relevant results. Numerical examples are also offered to confirm the theoretical findings. [ABSTRACT FROM AUTHOR]

We discrete the ergodic semilinear stochastic partial differential equations in space dimension d ≤ 3 with additive noise, spatially by a spectral Galerkin method and temporally by an exponential Euler scheme. It is shown that both the spatial semi-discretization and the spatio-temporal full discretization are ergodic. Further, convergence orders of the numerical invariant measures, depending on the regularity of noise, are recovered based on an easy time-independent weak error analysis without relying on Malliavin calculus. To be precise, the convergence order is 1 − ϵ in space and 1 2 − ϵ in time for the space-time white noise case and 2 − ϵ in space and 1 − ϵ in time for the trace class noise case in space dimension d = 1 , with arbitrarily small ϵ > 0. Numerical results are finally reported to confirm these theoretical findings. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This paper first establishes a fundamental mean-square convergence theorem for general one-step numerical approximations of Lévy noise driven stochastic differential equations with non-globally Lipschitz coefficients. Then two novel explicit schemes are designed and their convergence rates are exactly identified via the fundamental theorem. Different 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. However, we require that the Lévy measure is finite. 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+ht ∫Z N(ds,dz),t ≥ 0, h > 0 contribute to magnitude not more than O(h). Numerical results are finally reported to confirm the theoretical findings. [ABSTRACT FROM AUTHOR]

Abstract This paper aims to extend the scheme proposed in Jentzen et al. (2011) for stochastic partial differential equations (SPDEs) with global Lipschitz coefficients to non-global Lipschitz coefficients. We investigate the pathwise convergence of the scheme for a class of semilinear parabolic SPDEs with non-globally Lipschitz nonlinearity. We show first that the scheme is convergent uniformly in time and space under Lipschitz assumption on the nonlinearity of the SPDE, then we obtain the convergence in the case of non-globally Lipschitz nonlinearity via a localization technique. Compared to the scheme introduced in Jentzen (2009) for SPDEs under non-global Lipschitz coefficients, the scheme considered in this paper is simpler in the sense that the former uses two linear functionals of the noise while the latter uses one. [ABSTRACT FROM AUTHOR]

Abstract This paper examines the stability of numerical solutions of nonlinear stochastic differential equations (SDEs) with non-global Lipschitz continuous coefficients. Two implicit Milstein schemes, called drift-implicit Milstein scheme and double-implicit Milstein scheme, are considered to simulate the underlying SDEs. It is proved that the schemes can preserve the stability and contractivity in mean square of the underlying systems. [ABSTRACT FROM AUTHOR]

In order to simulate a linear stochastic oscillator with additive noise, improved nonstandard optimal (INSOPT) schemes are derived utilizing the nonstandard finite difference (NSFD) technique and the improvement technique. These proposed schemes reproduce long time features of the oscillator solution exactly. Their abilities in preserving the symplecticity, the linear growth property of the second moment and the oscillation property of the solution of the stochastic oscillator system on long time interval are studied. It can be shown that the component { xn } n≥1 of the INSOPT schemes switch signs infinitely many times as n - ∞ , almost surely. Further, the mean-square convergence order of 1 is obtained for these INSOPT schemes. Finally, numerical experiments illustrate intuitively the results obtained in this paper. [ABSTRACT FROM AUTHOR]

In this paper, a second-order backward differentiation formula compact difference scheme with the truncation error of orderfor time and 4 for space to fractional-order Volterra equation is considered. The integral term is treated by means of the second-order convolution quadrature suggested by Lubich and fourth-order accuracy compact approximation is applied for the second-order space derivative. The stability and convergence of the compact difference scheme in a new norm are proved by the energy method. Numerical experiments that are in total agreement with our analysis are reported. [ABSTRACT FROM AUTHOR]

In this paper, mean-square convergence and mean-square stability of θ -Maruyama methods are studied for nonlinear stochastic differential delay equations (SDDEs) with variable lag. Under global Lipschitz conditions, the methods are proved to be mean-square convergent with order 1 2 , and exponential mean-square stability of SDDEs implies that of the methods for sufficiently small step size h > 0 . Further, the exponential mean-square stability properties of SDDEs and those of numerical methods are investigated under some non-global Lipschitz conditions on the drift term. It is shown in this setting that the θ -Maruyama method with θ = 1 can preserve the exponential mean-square stability for any step size. Additionally, the θ -Maruyama method with 1 2 ≤ θ ≤ 1 is asymptotically mean-square stable for any step size, provided that the underlying system with constant lag is exponentially mean-square stable. Applications of this work to some special problem classes show that the results are deeper or sharper than those in the literature. Finally, numerical experiments are included to demonstrate the obtained theoretical results. [ABSTRACT FROM AUTHOR]