On discretization schemes for stochastic evolution equations
Istvan Gyöngy, Annie Millet
在巴拿赫空间中考虑了带单调算子的随机发展方程.给出了显式和隐式数值格式。证明了这些近似解的收敛性。
Stochastic evolutional equations with monotone operators are considered in Banach spaces. Explicit and implicit numerical schemes are presented. The convergence of the approximations to the solution of the equations is proved.
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On a space discretization scheme for the Fractional Stochastic Heat Equations
在这项工作中，我们引入了一个新的离散的分数拉普拉斯算子，并使用它来阐述一个近似计划的分数热方程由乘性圆柱白噪声扰动。特别是，我们估计的收敛速率。
On the discretization of backward doubly stochastic differential equations
本文讨论了倒向双随机微分方程的过程（Y，Z）解用前向过程X的逼近问题。在证明了Z的L ~ 2正则性之后，我们用Euler格式离散X，用张方法给出了过程（Y，Z）的离散格式。
Discretization Error of Stochastic Iterated Integrals
研究了在Skorohod意义下，当被积函数和积分函数为带跳半鞅时，随机迭代积分的离散化误差的弱收敛性.基于相关的归一化误差的渐近性质，研究了其逼近收敛速率，得到了当驱动过程是具有非零连续鞅分量的半鞅时，其逼近的收敛速度为$1/n$.作为应用，我们还研究了Doléans-Dade指数的离散化。
Weak convergence of error processes in discretizations of stochastic integrals and Besov spaces
考虑一类随机积分的Riemann离散所产生的重标误差过程的弱收敛性，并将弱极限的$L_p$ -可积性与随机积分的Malliavin意义下的分数阶光滑性联系起来.
Limit distribution of errors in discretization of stochastic Volterra equations with multidimensional kernel
本文研究了具有一般多维核结构的随机Volterra方程离散化误差的极限分布，将其推广至包含非分数型形式的对角矩阵核，并证明了重缩放误差过程的依律稳定收敛且在放宽假设下刻画了极限过程。
Error expansion for the discretization of Backward Stochastic Differential Equations
研究了一类解耦的正倒向随机微分方程$(X,Y,Z)$的时间离散误差。正向分量$X$是布朗随机微分方程的解，并且通过具有$N$时间步长的欧拉格式$X^N$来近似。后向分量由后向方案近似。首先，我们证明了在强$L\_p$意义下（$p \geq 1$）测量的误差$(Y^N-Y,Z^N-Z)$是$N^{-1/2}$阶的（这推广了Zhang 2004的结果）。其次，推导出误差展开式：令人惊讶的是，第一项与$X^N-X$成比例，而残差项的阶为$N^{-1}$。
Rate of convergence for discretization of integrals with respect to Fractional Brownian motion
本文考虑了一类凸函数$f$关于具有Hurst参数$H \in (1/2,1)$的分数布朗运动的随机积分$\int_{0}^{1} f'_-(B_t)\ud B_t$的一致离散化。在Statistics & Decisions，27，129-143中，对于任何凸函数$f$，证明了一致离散化对这样的随机积分的几乎必然收敛。本文证明了随机积分一致离散的$L^r$ -收敛性。此外，我们还得到了一个收敛速率。事实证明，可以使收敛的收敛速率尽可能接近$H - 1/2$。
Discretization error of Stochastic Integrals
研究了随机积分关于连续半鞅的一般随机剖分的黎曼和逼近的渐近误差分布。构造了渐近条件均值平方误差达到一个下界的有效离散格式。给出了两个应用实例;构造了有交易费用的有效Delta套期保值策略和Euler-Maruyama近似的有效离散化方案。
On implicit and explicit discretization schemes for parabolic SPDEs in any dimension
研究了空间相关高斯噪声扰动下任意维抛物型偏微分方程解$u(t,x)$的显式和隐式时空离散格式的收敛速度。这些系数仅依赖于$u(t,x)$，并且观察到相关性对速度的影响。
Central Limit Theorem for Irregular Discretization Scheme of Multilevel Monte Carlo Method
本文研究了多级蒙特卡洛方法不规则离散化方案的中心极限定理，得出了误差过程的渐近分布及Linderberg-Feller型定理。
An Implicit Euler Scheme with Non-uniform Time Discretization for Heat Equations with Multiplicative Noise
提出了一种求解随机热方程的算法，其关键是对驱动布朗运动$W$进行非均匀时间离散。对于这个算法，我们推导出一个误差界的$W$的一维组件的评价的数量。收敛速率取决于热方程的空间维数和$W$协方差的本征函数的衰减。根据已知的下界，我们的算法是最优的，直到一个常数，这种最优性不能通过均匀的时间离散化。
Hybrid stochastic differential equations are a useful tool to model continuously varying stochastic systems which are modulated by a random environment that may depend on the system state itself. In this paper, we establish the pathwise convergence of the solutions to hybrid stochastic differential equations via space-grid discretizations. While time-grid discretizations are a classical approach for simulation purposes, our space-grid discretization provides a link with multi-regime Markov modulated Brownian motions, leading to computational tractability. We exploit our convergence result to o…
Our study aims to specify the asymptotic error distribution in the discretization of a stochastic Volterra equation with a fractional kernel. It is well-known that for a standard stochastic differential equation, the discretization error, normalized with its rate of convergence $1/\sqrt{n}$, converges in law to the solution of a certain linear equation. Similarly to this, we show that a suitably normalized discretization error of the Volterra equation converges in law to the solution of a certain linear Volterra equation with the same fractional kernel.
Full Discretization of Stochastic Semilinear Schrödinger equation driven by multiplicative Wiener noise
本文针对乘性Wiener噪声驱动的随机半线性薛定谔方程，提出有限元与随机三角法结合的全离散格式，推导时空强收敛速率，并通过数值实验验证理论结果。
This paper studies the convergence of three temporal semi-discretizations for a backward semilinear stochastic evolution equation. For general terminal value and general coefficient with Lipschitz continuity, the convergence of the first two temporal semi-discretizations is established, and an explicit convergence rate is derived for the third temporal semi-discretization. The third temporal semi-discretization is applied to a general stochastic linear quadratic control problem, and the convergence of a temporally semi-discrete approximation to the optimal control is established.
Nonlinear stochastic evolution equations of second order with damping
证明了一类二阶非线性阻尼随机发展方程的全离散化的收敛性，从而证明了其解的存在性。离散格式结合了隐式时间步进格式和内部近似。唯一性也得到了证明。
Convergence of Finite Element Methods for Singular Stochastic Control
针对一类具有奇异性态的随机控制问题，提出了一种数值方法。这种方法解决了一个无限维的线性规划等价于随机控制问题，使用有限元型近似，这导致在一个可解的有限维程序。讨论了离散格式及必要的假设条件，并对离散格式进行了详细的收敛性分析。它的性能说明了两个例子，具有长期平均成本标准。
Euler time discretization of Backward Doubly SDEs and Application to Semilinear SPDEs
研究了一类半线性随机偏微分方程（简称SPDE）的数值概率解法。数值格式是基于解解耦的正倒向双随机微分方程系统的离散时间近似。在参数的标准假设下，证明了数值格式的收敛性和收敛速率。证明是基于向后方程的路径正则性结果的推广。
High order discretization schemes for stochastic volatility models
在通常的随机波动模型中，驱动资产价格波动的过程按照一个自治的一维随机微分方程演化。我们假设这个方程的系数是光滑的。利用伊藤公式，我们摆脱了在资产价格动态中，关于布朗运动驱动这个随机积分。利用这种结构，我们提出了一个计划，基于Milstein离散化的这个ESTA，与弱收敛的资产价格，一个计划，基于Ninomiya Victoir离散化的ESTA，与弱收敛的资产价格。我们还提出了一个具体的计划，提高了收敛性能时，资产价格的波动是由一个Orstein-Uhlenbeck过程。我们通过数值实验证实了理论上的收敛速度，并表明我们的方案很好地适应于Giles [2008 a，2008 b]引入的多层蒙特卡罗方法。
This paper investigates the influences of standard numerical discretizations on hitting probabilities for linear stochastic parabolic system driven by space-time white noises. We establish lower and upper bounds for hitting probabilities of the associated numerical solutions of both temporal and spatial semi-discretizations in terms of Bessel-Riesz capacity and Hausdorff measure, respectively. Moreover, the critical dimensions of both temporal and spatial semi-discretizations turn out to be half of those of the exact solution. This reveals that for a large class of Borel sets $A$, the probabil…
Long-time behavior for discretization schemes of Fokker-Planck equations via couplings
本文研究 Fokker-Planck 方程有限体积离散化对应马尔可夫链的长时间行为，利用耦合技术和路径耦合方法建立 $ϕ$-熵与 Wasserstein 距离的定量指数衰减，并进一步证明离散时间情形的指数平衡。
Ninomiya-Victoir scheme : strong convergence properties and discretization of the involved Ordinary Differential Equations
本文总结了前人关于Ninomiya-Victoir格式的强收敛速度及其归一化误差稳定收敛律的结果。然后，我们回顾的性质，多级蒙特卡罗估计涉及这个计划，我们介绍和研究之前。最后，我们对Ninomiya-Victoir格式中离散常微分方程所引入的误差进行了研究。证明了当采用4阶显式龙格-库塔法时，该误差强2阶收敛. 2)用于与布朗运动对应的常微分方程（分别为向量场。因此，我们放宽了Ninomiya和Ninomiya（2009）所需的布朗常微分方程的阶数5，以获得相同阶的强收敛性。此外，当使用这些龙格-库塔法时，我们的多层蒙特卡罗估计的性质被保留。
Time-discretization of stochastic 2-D Navier--Stokes equations with a penalty-projection method
本文研究了随机不可压缩纳维-斯托克斯方程的时间离散化问题，采用惩罚方法分析并推导了误差估计，最终得到速度和压力变量的主要算法在概率意义下的收敛速度为1/4阶，并利用全概率定律证明了该方案在强意义下的收敛性。
Stochastic evolution equations with compensated Poisson noise are considered in the variational approach with monotone and coercive coefficients. Here the Poisson noise is assumed to be time-homogeneous with $σ$-finite intensity measure on a metric space. By using finite element methods and Galerkin approximations, some explicit and implicit discretizations for this equation are presented and their convergence is proved. Polynomial growth condition and linear growth condition are assumed on the drift operator, respectively for the implicit and explicit schemes.
A high order time discretization of the solution of the non-linear filtering problem
连续时间滤波问题的解可以表示为信号过程中由观测路径参数化的某些泛函的两个期望之比，我们引入了一类任意阶泛函的离散化方案。所得结果推广了Picard的经典工作，Picard在滤波泛函中引入了一阶离散化.对于一个给定的时间间隔分区，我们构造离散化计划的收敛速度与$m$ -权力的网格的分区为任意$m\in\mathbb{N}$。这一结果为构造求解滤波问题的高阶数值逼近铺平了道路。
A Higher Order Discretization for the Stochastic Navier--Stokes equations with additive Noise
本文提出了一种针对带加性噪声的随机Navier-Stokes方程的高阶时间离散化方法，实现速度和压力在概率意义下的强收敛性。
A simple discretization scheme for nonnegative diffusion processes, with applications to option pricing
提出了非负扩散过程的一种离散格式，并利用鞅问题的框架证明了相应的近似过程序列的收敛性。这种方案的动机通常来自金融，特别是路径依赖的期权定价。该方案很简单：只要找到一个均值和方差满足一个简单条件的非负分布，那么，对于几乎任何（依赖于路径的）收益，由该方案得到的蒙特卡罗期权价格都将收敛于理论价格。模型和扩散过程的例子，该计划适用。
Efficient Discretization of Stochastic Integrals
给出了随机积分中离散化误差的期望二次变差的渐近下界。该理论依赖于一般随机变量的峰度和偏度的不等式，这些不等式本身似乎是新的。显式构造了达到下界的渐进有效方案。该结果直接适用于数学金融中的实际套期保值问题，给出了关于交易费用的再平衡日期和最优时间的渐近最优选择方法。渐近有效策略实际上反映了交易成本的结构。特别是一个特定的有偏再平衡计划被证明是优于无偏的计划，如果交易成本遵循凸模型。这个问题也讨论了指数效用最大化。
We prove that the implicit time Euler scheme coupled with finite elements space discretization for the 2D Navier-Stokes equations on the torus subject to a random perturbation converges in $L^2(Ω)$, and describe the rate of convergence for an $H^1$-valued initial condition. This refines previous results which only established the convergence in probability of these numerical approximations. Using exponential moment estimates of the solution of the stochastic Navier-Stokes equations and convergence of a localized scheme, we can prove strong convergence of this space-time approximation. The spee…
Long-time dynamics of stochastic wave equation with dissipative damping and its full discretization: exponential ergodicity and strong law of large numbers
针对非全局Lipschitz耗散阻尼的随机波动方程及其全离散化的长时间动力学研究空白，构造新型Lyapunov泛函证明其指数遍历性与唯一不变测度，给出不变测度误差估计并建立强大数定律。
In Becker and Jentzen (2019) and Becker et al. (2017), an explicit temporal semi-discretization scheme and a space-time full-discretization scheme were, respectively, introduced and analyzed for the additive noise-driven stochastic Allen-Cahn type equations, with strong convergence rates recovered. The present work aims to propose a different explicit full-discrete scheme to numerically solve the stochastic Allen-Cahn equation with cubic nonlinearity, perturbed by additive space-time white noise. The approximation is easily implementable, performing the spatial discretization by a spectral Gal…
Probabilistic limit behaviors of numerical discretizations for time-homogeneous Markov processes
本文研究时间齐次马尔可夫过程数值离散的时间平均估计量的概率极限行为，提出可验证的充分条件，适用于低正则性测试泛函，并涵盖多种随机系统。
We propose a new numerical method for one dimensional stochastic differential equations (SDEs). The main idea of this method is based on a representation of a weak solution of a SDE with a time changed Brownian motion, dated back to Doeblin (1940). In cases where the diffusion coefficient is bounded and $β$-Hölder continuous with $0 < β\leq 1$, we provide the rate of strong convergence. An advantage of our approach is that we approximate the weak solution, which enables us to treat a SDE with no strong solution. Our scheme is the first to achieve the strong convergence for the case $0 < β< 1/2…
Exact and high order discretization schemes for Wishart processes and their affine extensions
本文研究了正半定矩阵上的Wishart过程和仿射扩散问题。为此，我们专注于无穷小生成器的分裂，以便使用Ninomiya和Victoir或Alfonsi的合成技术。这样做，我们发现了一个显着的分裂Wishart过程，使我们能够准确地采样Wishart分布，没有任何限制的参数。它是相关的，但扩展了现有的精确模拟方法的基础上巴特利特的分解。此外，我们还可以构造Wishart过程的高阶离散格式和一般仿射扩散的二阶离散格式。这些方案实际上比精确模拟更快地对整个路径进行采样。给出了它们收敛性的数值结果。
Analysis of a HMM time-discretization scheme for a system of Stochastic PDE's
针对具有慢快分量的抛物型随机偏微分方程系统，提出一种基于平均原理的异构多尺度方法，采用两个耦合的半隐式欧拉格式离散时间，并推导了慢分量在强意义和弱意义下的误差界。
We define multiple stochastic integrals with respect to càdlàg martingales and prove moment bounds and chaos expansions, which allow to work with them in a way similar to Wiener stochastic integrals. In combination with the discretization framework of Erhard and Hairer (2017), our results give a tool for proving convergence of interacting particle systems to stochastic PDEs using regularity structures. As examples, we prove convergence of martingale-driven discretizations of the $3$-dimensional stochastic quantization equation and the KPZ equation.
Stochastic exponential integrators for finite element discretization of SPDEs for multiplicative and additive noise
针对乘性和加性噪声驱动的半线性抛物型随机偏微分方程，采用有限元空间离散和指数积分器时间离散，在迹类噪声假设下证明了均方L2范数收敛性，并通过二维线性反应扩散方程和非线性随机对流扩散反应方程验证了方法。
We discretize the stochastic Allen-Cahn equation with additive noise by means of a spectral Galerkin method in space and a tamed version of the exponential Euler method in time. The resulting error bounds are analyzed for the spatio-temporal full discretization in both strong and weak senses. Different from existing works, we develop a new and direct approach for the weak error analysis, which does not rely on the use of the associated Kolmogorov equation or Itô's formula and is therefore non-Markovian in nature. Such an approach thus has a potential to be applied to non-Markovian equations su…
Optimal order time discretizations for stochastic semilinear wave equations with multiplicative noise
本文提出两种带乘性噪声的随机半线性波方程的隐式时间离散方法，证明其能量稳定且收敛阶达到最优，并通过数值实验验证了理论误差估计的尖锐性。
Strong convergence rates for a full discretization of stochastic wave equation with nonlinear damping
本文研究一维和二维带非线性阻尼的随机波方程的时空全离散化，采用谱Galerkin方法和修正隐式指数Euler格式，克服超线性阻尼误差分析挑战，获得了均方收敛速率并经数值实验验证。
Semi-discretization for stochastic scalar conservation laws with multiple rough fluxes
本文提出半离散化方法用于求解具有多粗糙时间依赖的非均匀通量的随机标量守恒律，基于Brenier运输-坍缩算法和粗糙路径定义的特征线，证明了强L^1收敛性及收敛速度。
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。
In this survey we present the near-optimal stochastic control problem according to some recent tools in the literature. In particular, we focus on the approach of a discretization of the noise values instead of the canonical time-discretization. This is the so called {\it skeleton} structure. This allows to obtain an $ε$-optimal control in non-Markovian systems (the main Theorem). A simple example illustrates the technique. The importance of the approach is emphasised in a final section on open problems related to more geometrical framework and discontinuous noise.
Many physical, chemical and biological systems have an inherent discrete spatial structure that strongly influences their dynamical behaviour. Similar remarks apply to internal or external noise, as well as to nonlocal coupling. In this paper we study the combined effect of nonlocal spatial discretization and stochastic perturbations on travelling waves in the Nagumo equation, which is a prototypical model for bistable reaction-diffusion partial differential equations (PDEs). We prove that under suitable parameter conditions, various discrete-stochastic variants of the Nagumo equation have sol…
Approximation of the invariant law of SPDEs: error analysis using a Poisson equation for a full-discretization scheme
本文研究了加性空间时间白噪声下全离散非线性随机偏微分方程的长期行为，证明了其不变测度的近似值与真实测度的收敛率。
Strong convergence of finite element approximations for a fourth-order stochastic pseudo-parabolic equation with additive noise
本文研究了四阶随机伪抛物方程在加性噪声下的有限元近似强收敛性，采用有限元和半隐式方法，验证了理论收敛率。
An exponential integrator scheme for time discretization of nonlinear stochastic wave equation
本文研究了非线性随机波动方程时间离散中指数积分格式的收敛性分析，提出了一种统一框架，涵盖加性与乘性噪声的重要情况，并证明了该格式在强$L^p$意义下的一致收敛性及精确收敛率，进一步分析了加性噪声下的弱收敛率，不依赖 Malliavin 计算，展示了该方法比隐式欧拉和 Crank-Nicolson 时间离散具有更高的收敛率。
We consider the strong solution of the 2D Navier-Stokes equations in a torus subject to an additive noise. We implement a fully implicit time numerical scheme and a finite element method in space. We prove that the rate of convergence of the schemes is $η\in[0,1/2)$ in time and 1 in space. Let us mention that the coefficient $η$ is equal to the time regularity of the solution with values in $\LL^2$. Our method relies on the existence of finite exponential moments for both the solution and its time approximation. Our main idea is to use a discrete Gronwall lemma for the error estimate without a…
Stability of gas measures under perturbations and discretizations
对于一般的气体模型--包括离散的和连续的Gibbsian模型以及轮廓或聚合物系综--我们确定了一个\emph{diluteness condition}，它意味着：（1）无限体积平衡测度的唯一性;（2）该测度在参数摄动和离散化方案下的稳定性，（3）无限体积测度及其扰动和离散的耦合完美模拟格式的存在性。这些结果中的一些先前已经通过基于聚类扩展的方法获得。相比之下，我们的治疗是纯粹的概率和稀释条件是弱于现有的集群扩张的收敛条件。
Analysis of kinetic Langevin Monte Carlo under the stochastic exponential Euler discretization from underdamped all the way to overdamped
本文研究了基于随机指数欧拉离散化的动能朗之万蒙特卡洛方法，分析了在不同参数选择下算法的稳定性与收敛性，得出了在过阻尼情况下算法仍能稳定运行的结论。
The primary emphasis of this work is the development of a finite element based space-time discretization for solving the stochastic Lagrangian averaged Navier-Stokes (LANS-$α$) equations of incompressible fluid turbulence with multiplicative random forcing, under nonperiodic boundary conditions within a bounded polygonal (or polyhedral) domain of R^d , d $\in$ {2, 3}. The convergence analysis of a fully discretized numerical scheme is investigated and split into two cases according to the spacial scale $α$, namely we first assume $α$ to be controlled by the step size of the space discretizatio…
Finite Element Approximations of Stochastic Linear Schrödinger equation driven by additive Wiener noise
本文研究了加性维纳噪声驱动下随机线性薛定谔方程的有限元近似方法，推导了误差估计并进行了数值验证。
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