Euler scheme for density dependent stochastic differential equations

A note on Euler approximations for stochastic differential equations with delay

An existence and uniqueness theorem for a class of stochastic delay differential equations is presented, and the convergence of Euler approximations for these equations is proved under general conditions. Moreover, the rate of almost sure convergence is obtained under local Lipschitz and also under monotonicity conditions.

Error Distribution Of The Euler Approximation Scheme For Stochastic Volterra Integral Equations

The purpose of this paper is to establish the convergence in distribution of the normalized error in the Euler approximation scheme for stochastic Volterra equations driven by a standard Brownian motion, with a kernel of the form $(t-s)^α$, where $α\in (-\frac 12, \frac 12)$.

Stationary Solutions of Damped Stochastic 2-dimensional Euler's Equation

Existence of stationary point vortices solution to the damped and stochastically driven Euler's equation on the two dimensional torus is proved, by taking limits of solutions with finitely many vortices. A central limit scaling is used to show in a similar manner the existence of stationary solutions with white noise marginals.

Rate of convergence of Euler approximations of solution to mixed stochastic differential equation involving Brownian motion and fractional Brownian motion

We consider a mixed stochastic differential equation involving both standard Brownian motion and fractional Brownian motion with Hurst parameter $H>1/2$. The mean-square rate of convergence of Euler approximations of solution to this equation is obtained.

Scaling limit of stochastic 2D Euler equations with transport noises to the deterministic Navier-Stokes equations

We consider a family of stochastic 2D Euler equations in vorticity form on the torus, with transport type noises and $L^2$-initial data. Under a suitable scaling of the noises, we show that the solutions converge weakly to that of the deterministic 2D Navier--Stokes equations. Consequently, we deduce that the weak solutions of the stochastic 2D Euler equations are approximately unique and "weakly quenched exponential mixing".

On a 2D stochastic Euler equation of transport type: existence and geometric formulation

We prove weak existence of Euler equation (or Navier-Stokes equation) perturbed by a multiplicative noise on bounded domains of $\mathbb R^2$ with Dirichlet boundary conditions and with periodic boundary conditions. Solutions are $H^1$ regular. The equations are of transport type.

On the Euler-Maruyama approximation for one-dimensional stochastic differential equations with irregular coefficients

We study the strong rates of the Euler-Maruyama approximation for one dimensional stochastic differential equations whose drift coefficient may be neither continuous nor one-sided Lipschitz and diffusion coefficient is Hölder continuous. Especially, we show that the strong rate of the Euler-Maruyama approximation is 1/2 for a large class of equations whose drift is not continuous. We also provide the strong rate for equations whose drift is Hölder continuous and diffusion is nonconstant

Strong Convergence of Euler Approximations of Stochastic Differential Equations with Delay under Local Lipschitz Condition

The strong convergence of Euler approximations of stochastic delay differential equations is proved under general conditions. The assumptions on drift and diffusion coefficients have been relaxed to include polynomial growth and only continuity in the arguments corresponding to delays. Furthermore, the rate of convergence is obtained under one-sided and polynomial Lipschitz conditions. Finally, our findings are demonstrated with the help of numerical simulations.

Kolmogorov equations associated to the stochastic 2D Euler equations

The Kolmogorov equation associated to a stochastic 2D Euler equations with transport type noise and random initial conditions is studied by a direct approach, based on Fourier analysis, Galerkin approximation and Wiener chaos methods. The method allows us to generalize previous results and to understand the role of the regularity of the noise, in relation to a limiting value of roughness.

On the Euler--Maruyama scheme for degenerate stochastic differential equations with non-sticky condition

The aim of this paper is to study weak and strong convergence of the Euler--Maruyama scheme for a solution of one-dimensional degenerate stochastic differential equation $\mathrm{d} X_t=σ(X_t) \mathrm{d} W_t$ with non-sticky condition. For proving this, we first prove that the Euler--Maruyama scheme also satisfies non-sticky condition. As an example, we consider stochastic differential equation $\mathrm{d} X_t=|X_t|^α \mathrm{d} W_t$, $α\in (0,1/2)$ with non-sticky boundary condition and we give some remarks on CEV models in mathematical finance.

欧拉-马尔可夫方法用于乘性分数布朗运动驱动的分布依赖随机微分方程

Euler-Maruyama method for distribution dependent stochastic differential equation driven by multiplicative fractional Brownian motion

本文研究了由乘性分数布朗运动驱动的分布依赖随机微分方程的欧拉-马尔可夫方法，推导了误差上界并验证了其正确性。

带分数核的随机Volterra方程的多层Monte Carlo Euler估计量的中心极限定理

Central limit theorem of Multilevel Monte Carlo Euler estimators for Stochastic Volterra equations with fractional kernels

本文证明了带分数核的随机Volterra方程在Euler离散化下多层Monte Carlo估计量满足Lindeberg-Feller型中心极限定理。

$G$-框架下随机微分方程、随机微分延迟方程及其相应 Euler-Maruyama 方法的稳定性等价性

Stability equivalence for stochastic differential equations, stochastic differential delay equations and their corresponding Euler-Maruyama methods in $G$-framework

该研究证明在 $G$-框架下，当延迟或数值步长足够小时，$G$-SDDE、辅助 $G$-SDE 及其 EM 方法在 $p$ 阶矩实际指数稳定性上彼此等价，从而支持通过模拟检验稳定性。

Modified Euler approximation scheme for stochastic differential equations driven by fractional Brownian motions

For a stochastic differential equation driven by a fractional Brownian motion with Hurst parameter $H> \frac12$ it is known that the classical Euler scheme has the rate of convergence $2H-1$. In this paper we introduce a new numerical scheme which is closer to the classical Euler scheme for diffusion processes, in the sense that it has the rate of convergence $2H-\frac12$. In particular, the rate of convergence becomes $\frac 12$ when $H$ is formally set to $\frac 12$ (the rate of Euler scheme for classical Brownian motion). The rate of weak convergence is also deduced for this scheme. The mai…

Euler-Lagrangian approach to stochastic Euler equations in Sobolev Spaces

The purpose of this paper is to establish the equivalence between Lagrangian and classical formulations for the stochastic incompressible Euler equations, the proof is based in Ito-Wentzell-Kunita formula and stochastic analysis techniques. Moreover, we prove a local existence result for the Lagrangian formulation in suitable Sobolev Spaces.

麦肯-瓦拉索夫随机微分方程不变测度的欧拉-马约拉纳方法

The Euler-Maruyama method for invariant measures of McKean-Vlasov stochastic differential equations

本文研究了在单调性条件下，使用欧拉-马约拉纳方法近似麦肯-瓦拉索夫随机微分方程的不变测度，证明了数值解的收敛性及唯一不变测度的收敛率。

Stochastic differential equations driven by fractional Brownian motion with locally Lipschitiz drift and their Euler approximation

In this paper, we study a class of one-dimensional stochastic differential equations driven by fractional Brownian motion with Hurst parameter $H>\ff 1 2$. The drift term of the equation is locally Lipschitz and unbounded in the neighborhood of $0$. We show the existence, uniqueness and positivity of the solutions. The estimations of moments, including the negative power moments, are given. Based on these estimations, strong convergence of the positivity preserving drift-implicit Euler-type scheme is proved, and optimal convergence rate is obtained. By using Lamperti transformation, we show th…

Invariant measures for stochastic damped 2D Euler equations

We study the two-dimensional Euler equations, damped by a linear term and driven by an additive noise. The existence of weak solutions has already been studied; pathwise uniqueness is known for solutions that have vorticity in $L^\infty$. In this paper, we prove the Markov property and then the existence of an invariant measure in the space $L^\infty$ by means of a Krylov-Bogoliubov's type method, working with the weak$\star$ and the bounded weak$\star$ topologies in $L^\infty$.

The rate of convergence of Euler approximations for solutions of stochastic differential equations driven by fractional Brownian motion

The paper focuses on discrete-type approximations of solutions to non-homogeneous stochastic differential equations (SDEs) involving fractional Brownian motion (fBm). We prove that the rate of convergence for Euler approximations of solutions of pathwise SDEs driven by fBm with Hurst index $H>1/2$ can be estimated by $O(δ^{2H-1})$ ($δ$ is the diameter of partition). For discrete-time approximations of Skorohod-type quasilinear equation driven by fBm we prove that the rate of convergence is $O(δ^H)$.

Strong convergence of a half-explicit Euler scheme for constrained stochastic mechanical systems

This paper is concerned with the numerical approximation of stochastic mechanical systems with nonlinear holonomic constraints. Such systems are described by second order stochastic differential-algebraic equations involving an implicitly given Lagrange multiplier process. The explicit representation of the Lagrange multiplier leads to an underlying stochastic ordinary differential equation, the drift coefficient of which is typically not globally one-sided Lipschitz continuous. We investigate a half-explicit drift-truncated Euler scheme which fulfills the constraint exactly. Pathwise uniform …

带Kraichnan噪声和$L^p$数据的随机二维Euler方程的路径唯一性

On the pathwise uniqueness of stochastic 2D Euler equations with Kraichnan noise and $L^p$-data

本文将随机二维Euler方程涡量形式解的路径唯一性从$p>3/2$推广到任意$p>1$，去除了Coghi和Maurelli结果中对$L^p$指数的限制。

SU(N)上快速-慢速随机欧拉-阿诺德系统中的余伴随平均和跨尺度通量

Coadjoint averaging and cross-scale fluxes in a fast-slow stochastic Euler-Arnold system on $SU(N)$

研究 Zeitlin 的 SU(N) 近似随机版中能量和涡度相关跨尺度通量，引入快速-慢速随机框架，表明余伴随对称性抑制平均非线性通量，其打破可选择跨尺度转移方向。

Speed of convergence of time Euler schemes for a stochastic 2D Boussinesq model

We prove that an implicit time Euler scheme for the 2D-Boussinesq model on the torus $D$ converges. Various moment of the $W^{1,2}$-norms of the velocity and temperature, as well as their discretizations, are computed. We obtain the optimal speed of convergence in probability, and a logarithmic speed of convergence in $L^2(Ω)$. These results are deduced from a time regularity of the solution both in $L^2(D)$ and $W^{1,2}(D)$, and from an $L^2(Ω)$ convergence restricted to a subset where the $W^{1,2}$-noms of the solutions are bounded.

On a Stochastic Leray-α model of Euler equations

We deal with the 3D inviscid Leray-α model. The well posedness for this problem is not known; by adding a random perturbation we prove that there exists a unique (in law) global solution. The random forcing term formally preserves conservation of energy. The result holds for initial velocity of finite energy and the solution has finite energy a.s.. These results are easily extended to the 2D case.

Unique Ergodicity for Fractionally Dissipated, Stochastically Forced 2D Euler Equations

We establish the existence and uniqueness of an ergodic invariant measure for 2D fractionally dissipated stochastic Euler equations on the periodic box, for any power of the dissipation term.

A transformed stochastic Euler scheme for multidimensional transmission PDE

In this paper we consider multi-dimensional partial differential equations of parabolic type involving divergence form operators that possess a discontinuous coefficient matrix along some smooth interface. The solution of the equation is assumed to present a compatibility transmission condition of its conormal derivatives at this interface (multi-dimensional diffraction problem). We prove an existence and uniqueness result for the solution and construct a low complexity numerical Monte Carlo stochastic Euler scheme to approximate the solution of the parabolic partial differential equation in d…

Convex ordering for stochastic Volterra equations and their Euler schemes

In this paper, we are interested in comparing solutions to stochastic Volterra equations for the convex order on the space of continuous $\R^d$-valued paths and for the monotonic convex order when $d=1$. Even if in general these solutions are neither semi-martingales nor Markov processes, we are able to exhibit conditions on their coefficients enabling the comparison. Our approach consists in first comparing their Euler schemes and then taking the limit as the time step vanishes. We consider two types of Euler schemes depending on the way the Volterra kernels are discretized. The conditions en…

Convergence Rate of Euler-Maruyama Scheme for SDEs with Rough Coefficients

In this paper, we are concerned with convergence rate of Euler-Maruyama scheme for stochastic differential equations with rough coefficients. The key contributions lie in (i), by means of regularity of non-degenerate Kolmogrov equation, we investigate convergence rate of Euler-Maruyama scheme for a class of stochastic differential equations, which allow the drifts to be Dini-continuous and unbounded; (ii) by the aid of regularization properties of degenerate Kolmogrov equation, we discuss convergence rate of Euler-Maruyama scheme for a range of degenerate stochastic differential equations, whe…

Stochastic isentropic Euler equations

We study the stochastically forced system of isentropic Euler equations of gas dynamics with a $γ$-law for the pressure. We show the existence of martingale weak entropy solutions; we also discuss the existence and characterization of invariant measures in the concluding section.

Energy-balance for the incompressible Euler equations with stochastic forcing

We establish energy-balance for weak solutions of the stochastically forced incompressible Euler equations, enjoying Hölder regularity $C^α$, $α>1/3$. It is well known as the Onsager's conjecture for the deterministic incompressible Euler equations, which describes the energy conservation of weak solutions having Hölder regularity $C^α$, $α>1/3$. Additionally, we obtain energy-balance for the inhomogeneous incompressible Euler system driven by a cylindrical Wiener process.

分数布朗运动驱动的随机函数微分方程的欧拉方案

Euler scheme for stochastic functional differential equations driven by fractional Brownian motion

本文利用粗糙路径技术，研究分数布朗运动驱动的随机函数微分方程的近似解，采用Young积分并基于λ-Holder连续函数系数，提出收敛率为1/n^γ的欧拉方案，并通过数值模拟验证理论结果。

2D Euler equations with Stratonovich transport noise as a large scale stochastic model reduction

The limit from an Euler type system to the 2D Euler equations with Stratonovich transport noise is investigated. A weak convergence result for the vorticity field and a strong convergence result for the velocity field are proved. Our results aim to provide a stochastic reduction of fluid-dynamics models with three different time scales.

通过非线性Itô噪声实现Euler方程的无爆破

No blow-up by nonlinear Itô noise for the Euler equations

本文通过引入具径向结构且超线性增长的乘性Itô噪声，证明二维和三维随机Euler方程存在唯一整体强解，并推广至一类SPDEs，证明此类噪声能有效防止解的爆破。

Dirichlet forms methods, an application to the propagation of the error due to the Euler scheme

We present recent advances on Dirichlet forms methods either to extend financial models beyond the usual stochastic calculus or to study stochastic models with less classical tools. In this spirit, we interpret the asymptotic error on the solution of an sde due to the Euler scheme in terms of a Dirichlet form on the Wiener space, what allows to propagate this error thanks to functional calculus.

从随机哈密顿系统到随机可压缩 Euler 方程

From Stochastic Hamiltonian Systems to Stochastic Compressible Euler Equation

本文研究通过长程势相互作用的 $N$ 粒子随机哈密顿系统，证明其位置与速度经验测度在粒子数趋于无穷时收敛到随机可压缩 Euler 方程的解，并以 Sobolev 范数量化收敛距离。

Consistency and Asymptotic Normality of Stochastic Euler Schemes for Ordinary Differential Equations

General stochastic Euler schemes for ordinary differential equations are studied. We give proofs on the consistency, the rate of convergence and the asymptotic normality of these procedures.

On stochastic Euler-Poincaré equations driven by pseudo-differential/multiplicative noise

In this paper we focus on the stochastic Euler-Poincaré equations with pseudo-differential/multiplicative noise. We first establish two new cancellation properties on pseudo-differential operators, which play a key role in energy estimate. Then, we obtain results on local solution, blow-up criterion and global existence. The interplay between stability on exiting times and continuous dependence of solution on initial data are also studied for the multiplicative noise case.

Uniqueness and energy balance for isentropic Euler equation with stochastic forcing

In this article, we prove uniqueness and energy balance for isentropic Euler system driven by a cylindrical Wiener process. Pathwise uniqueness result is obtained for weak solutions having Hölder regularity $C^α,α>1/2$ in space and satisfying one-sided Lipschitz bound on velocity. We prove Onsager's conjecture for isentropic Euler system with stochastic forcing, that is, energy balance equation for solutions enjoying Hölder regularity $C^α,α>1/3$. Both the results have been obtained in a more general setting by considering regularity in Besov space.

Euler and Betti curves are stable under Wasserstein deformations of distributions of stochastic processes

Euler and Betti curves of stochastic processes defined on a $d$-dimensional compact Riemannian manifold which are almost surely in a Sobolev space $W^{n,s}(X,\mathbb{R})$ (with $d<n$) are stable under perturbations of the distributions of said processes in a Wasserstein metric. Moreover, Wasserstein stability is shown to hold for all $p>\frac{d}{n}$ for persistence diagrams stemming from functions in $W^{n,s}(X,\mathbb{R})$.

Hölder空间中随机三维Euler方程的平稳解

Stationary solutions to stochastic 3D Euler equations in Hölder space

本研究提出新型随机凸积分方法，结合已有随机凸积分技术与路径估计，证明加性噪声驱动的三维Euler方程在特定Hölder空间中存在无穷多全局平稳解。

Full well-posedness of point vortex dynamics corresponding to stochastic 2D Euler equations

The motion of a finite number of point vortices on a two-dimensional periodic domain is considered. In the deterministic case it is known to be well posed only for almost every initial configuration. Coalescence of vortices may occur for certain initial conditions. We prove that when a generic stochastic perturbation compatible with the Eulerian description is introduced, the point vortex motion becomes well posed for every initial configuration, in particular coalescence disappears.

Stochastic compressible Euler equations and inviscid limits

We prove the existence of a unique local strong solution to the stochastic compressible Euler system with nonlinear multiplicative noise. This solution exists up to a positive stopping time and is strong in both the PDE and probabilistic sense. Based on this existence result, we study the inviscid limit of the stochastic compressible Navier--Stokes system. As the viscosity tends to zero, any sequence of finite energy weak martingale solutions converges to the compressible Euler system.

Strong Rate of Convergence for the Euler-Maruyama Approximation of Stochastic Differential Equations with Irregular Coefficients

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 one-sided Lipschitz condition and the diffusion coefficient is Hölder continuous.

Strong convergence rate of the truncated Euler-Maruyama method for stochastic differential delay equations with Poisson jumps

In this paper, we study a class of super-linear stochastic differential delay equations with Poisson jumps (SDDEwPJs). The convergence and rate of the convergence of the truncated Euler-Maruyama numerical solutions to SDDEwPJs are investigated under the generalized Khasminskii-type condition.

Existence for stochastic 2D Euler equations with positive $H^{-1}$ vorticity

We prove the existence of non-negative measure- and $H^{-1}$-valued vorticity solutions to the stochastic 2D Euler equations with transport vorticity noise, starting from any non-negative vortex sheet. This extends the result by Delort (J.Amer.Math.Soc. 1991) to the stochastic case.

带随机强迫的不可压缩欧拉方程：一种几何方法

Incompressible Euler equations with stochastic forcing: a geometric approach

本文采用无限维几何方法研究带随机强迫的欧拉-阿诺德方程，在Sobolev空间中建立了强解的局部存在唯一性，为随机欧拉方程的适定性提供了新工具。

On ill- and well-posedness of dissipative martingale solutions to stochastic 3D Euler equations

We are concerned with the question of well-posedness of stochastic three dimensional incompressible Euler equations. In particular, we introduce a novel class of dissipative solutions and show that (i) existence; (ii) weak--strong uniqueness; (iii) non-uniqueness in law; (iv) existence of a strong Markov solution; (v) non-uniqueness of strong Markov solutions; all hold true within this class. Moreover, as a byproduct of (iii) we obtain existence and non-uniqueness of probabilistically strong and analytically weak solutions defined up to a stopping time and satisfying an energy inequality.

通过随机凸积分求解随机三维欧拉方程的Hölder连续解

Hölder continuous solutions to stochastic 3D Euler equations via stochastic convex integration

该研究针对加法随机外力驱动的三维欧拉方程，采用随机凸积分方法构造了全局Hölder连续平稳解并得到一致矩估计，同时证明了任意无散Hölder初值下无穷多全局概率强解析弱解的存在性。

Dissipative solutions and Markov selection to the complete stochastic Euler system

We introduce the concept of stochastic measure-valued solutions to the complete Euler system describing the motion of a compressible inviscid fluid subject to stochastic forcing, where the nonlinear terms are described by defect measures. These solutions are weak in the probabilistic sense (probability space is not a given `priori', but part of the solution) and analytical sense (derivatives only exists in the sense distributions). In particular, we show that: existence, weak-strong principle; a weak measure-valued solution coincides with a strong solution provided the later exists, all hold t…

Existence and uniqueness for stochastic 2D Euler flows with bounded vorticity

Strong existence and pathwise uniqueness of solutions with $L^{\infty}$-vorticity of 2D stochastic Euler equations is proved. The noise is multiplicative and involves first derivatives. A Lagrangian approach is implemented, where a stochastic flow solving a nonlinear flow equation is constructed. Stability under regularization is also proved.

tamed Euler schemes 对于一类随机演变方程的收敛性

Convergence of tamed Euler schemes for a class of stochastic evolution equations

本文研究了一类具有超线性增长算子的随机演变方程的稳定性与收敛性，采用tamed Euler方法结合Galerkin方案进行时空离散化。

一维随机微分方程中局部时间的欧拉-马尔可夫 approximation 的弱收敛率

The weak rate of convergence for the Euler-Maruyama approximation of one-dimensional stochastic differential equations involving the local times of the unknown process

本文研究了一维随机微分方程中局部时间的欧拉-马尔可夫 approximation 的弱收敛率，通过变换去除局部时间，推导了无局部时间的近似方法，并给出了任意函数G在特定类中的弱收敛率。

随机时间步长欧拉方法及其连续动力学

The random timestep Euler method and its continuous dynamics

本文提出随机欧拉动力学这一连续时间马尔可夫过程，用于刻画随机时间步长欧拉方法，并通过无穷小生成元分析、局部截断误差分析和Foster-Lyapunov准则证明了其收敛性与稳定性。

Local and global existence of smooth solutions for the stochastic Euler equations with multiplicative noise

We establish the local existence of pathwise solutions for the stochastic Euler equations in a three-dimensional bounded domain with slip boundary conditions and a very general nonlinear multiplicative noise. In the two-dimensional case we obtain the global existence of these solutions with additive or linear-multiplicative noise. Lastly, we show that, in the three dimensional case, the addition of linear multiplicative noise provides a regularizing effect; the global existence of solutions occurs with high probability if the initial data is sufficiently small, or if the noise coefficient is s…