Related papers: Euler Estimates of Rough Differential Equations
We construct a nonstandard finite difference numerical scheme to approximate stochastic differential equations (SDEs) using the idea of weighed step introduced by R.E. Mickens. We prove the strong convergence of our scheme under locally…
The Euler scheme is a standard time discretization for BSDEs, but its implementation hinges on approximating conditional expectations and the associated martingale terms at each time step. We propose an implementation based on the Wiener…
The paper studies the rate of convergence of the weak Euler approximation for solutions to possibly completely degenerate SDEs driven by Levy processes, with Hoelder-continuous coefficients. It investigates the dependence of the rate on the…
We deal with approximation of solutions of delay differential equations (DDEs) via the classical Euler algorithm. We investigate the pointwise error of the Euler scheme under nonstandard assumptions imposed on the right-hand side function…
The present paper is devoted to the study of sample paths of G-Brownian motion and stochastic differential equations (SDEs) driven by G-Brownian motion from the view of rough path theory. As the starting point, we show that quasi-surely,…
In the present work, we delve into further study of numerical approximations of SDEs with non-globally monotone coefficients. We design and analyze a new family of stopped increment-tamed time discretization schemes of Euler, Milstein and…
We investigate existence, uniqueness and approximation of solutions to stochastic delay differential equations (SDDEs) under Carath\'eodory-type drift coefficients. Moreover, we also assume that both drift $f=f(t,x,z)$ and diffusion…
This work develops moment bounds for the controlled rough path norm of the solution of semilinear rough partial differential equations.~The novel aspects are two-fold: first we consider rough paths of low time regularity…
We introduce deterministic perturbation schemes for the recently proposed random directions stochastic approximation (RDSA) [17], and propose new first-order and second-order algorithms. In the latter case, these are the first second-order…
Recently, it has been shown in [Hairer, M., Hutzenthaler, M., Jentzen, A., Loss of regularity for Kolmogorov equations, Ann. Probab. 43, 2 (2015), 468--527] that there exists a system of stochastic differential equations (SDE) on the time…
We provide a unified analytic approach to study stationary states of controlled differential equations driven by rough paths, using the framework of random dynamical systems and random attractors. Part I deals with driving paths of finite…
We study the Euler scheme for a stochastic differential equation driven by a Levy process Y. More precisely, we look at the asymptotic behavior of the normalized error process u_n(X^n-X), where X is the true solution and X^n is its Euler…
In this paper we consider the numerical approximation of a general second order semi-linear parabolic partial differential equation. Equations of this type arise in many contexts, such as transport in porous media. Using finite element…
Our subject of study is strong approximation of stochastic differential equations (SDEs) with respect to the supremum error criterion, and we seek approximations that are strongly asymptotically optimal in specific classes of…
In many recent applications when new materials and technologies are developed it is important to describe and simulate new nonlinear and nonlocal diffusion transport processes. A general class of such models deals with nonlocal fractional…
We provide a general method to analyze the asymptotic properties of a variety of estimators of continuous time diffusion processes when the data are not only discretely sampled in time but the time separating successive observations may…
We are interested in the kernel of one-dimensional diffusion equations with continuous coefficients as evaluated by means of explicit discretization schemes of uniform step $h>0$ in the limit as $h\to0$. We consider both semidiscrete…
We discretize a risk-neutral optimal control problem governed by a linear elliptic partial differential equation with random inputs using a Monte Carlo sample-based approximation and a finite element discretization, yielding finite…
This paper considers the implicit Euler discretization of Levant's arbitrary order robust exact differentiator in presence of sampled measurements. Existing implicit discretizations of that differentiator are shown to exhibit either…
Based on two isomorphisms of Hopf algebras, we provide a bound in the optimal order on the remainder of the truncated Taylor expansion for controlled differential equations driven by branched rough paths.