Related papers: Exponential integrability properties of numerical …
This work focuses on the numerical approximations of neutral stochastic delay differential equations with their drift and diffusion coefficients growing super-linearly with respect to both delay variables and state variables. Under…
This paper aims at developing a systematic study for the weak rate of convergence of the Euler-Maruyama scheme for stochastic differential equations with very irregular drift and constant diffusion coefficients. We apply our method to…
We consider numerical approximations of stochastic differential equations by the Euler method. In the case where the SDE is elliptic or hypoelliptic, we show a weak backward error analysis result in the sense that the generator associated…
In this paper we study a type of stochastic McKean-Vlasov equations with non-Lipschitz coefficients. Firstly, by an Euler-Maruyama approximation existence of its weak solutions is proved. And then we observe pathwise uniqueness of its weak…
In this paper a variant of nonlinear exponential Euler scheme is proposed for solving nonlinear heat conduction problems. The method is based on nonlinear iterations where at each iteration a linear initial-value problem has to be solved.…
Polynomial stability of exact solution and modified truncated Euler-Maruyama method for stochastic differential equations with time-dependent delay are investigated in this paper. By using the well known discrete semimartingale convergence…
We consider the Euler-Maruyama approximation for multi-dimensional stochastic differential equations with irregular coefficients. We provide the rate of strong convergence where the possibly discontinuous drift coefficient satisfies a…
We consider one-step methods for integrating stochastic differential equations and prove pathwise convergence using ideas from rough path theory. In contrast to alternative theories of pathwise convergence, no knowledge is required of…
We study the accuracy of the expected Euler characteristic approximation to the distribution of the maximum of a smooth, centered, unit variance Gaussian process f. Using a point process representation of the error, valid for arbitrary…
A quantitative definition of numerical stiffness for initial value problems is proposed. Exponential integrators can effectively integrate linearly stiff systems, but they become expensive when the linear coefficient is a matrix, especially…
We prove several properties of the EMCEL scheme, which is capable of approximating one-dimensional continuous strong Markov processes in distribution on the path space (the scheme is briefly recalled). Special cases include irregular…
Using lattice approximations of Euclidean space, we develop a way to approximate stable processes that are represented by stochastic integrals over Euclidean space. Via a stable version of the Lindeberg-Feller Theorem we show that the…
We consider the approximation of stochastic differential equations (SDEs) with non-Lipschitz drift or diffusion coefficients. We present a modified explicit Euler-Maruyama discretisation scheme that allows us to prove strong convergence,…
We consider the Euler scheme for stochastic differential equations with jumps, whose intensity might be infinite and the jump structure may depend on the position. This general type of SDE is explicitly given for Feller processes and a…
In this paper, we consider a "compensated" random sum that arises from numerical approximation of stochastic integrations and differential equations. We show that the compensated sum exhibits some surprising cancellations among its…
This paper addresses the challenging numerical simulation of nonlinear hybrid stochastic functional differential equations with infinite delays. We first propose an explicit scheme using space and time truncation, requiring only finite…
We present the singular Euler--Maclaurin expansion, a new method for the efficient computation of large singular sums that appear in long-range interacting systems in condensed matter and quantum physics. In contrast to the traditional…
In this paper, we present new types of exponential integrators for Stochastic Differential Equations (SDEs) that take the advantage of the exact solution of (generalised) geometric Brownian motion. We examine both Euler and Milstein…
In this paper we present some open problems pertaining to the approximation theory involved in the solution of the important class of Nonlinear Partial Differential Equations (NPDEs) of integrable type. For this class of NPDEs, any Initial…
Stiff ordinary differential equations (ODEs) are common in many science and engineering fields, but standard neural ODE approaches struggle to accurately learn these stiff systems, posing a significant barrier to widespread adoption of…