Tetyana Kosenkova, Alexei Kulik, Ilya Pavlyukevich
For solutions $X=(X_t)_{t\in[0,T]}$ of Lévy-driven Marcus stochastic differential equations we study the Wong--Zakai type time discrete approximations $\bar X=(\bar X_{kh})_{0\leq k\leq T/h}$, $h>0$, and establish the first order convergence $|E f(X_T)-E f(X^h_T)|\leq C h$ for $f\in C_b^4$.
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In this paper we obtain a Wong-Zakai approximation to solutions of backward doubly stochastic differential equations.
In this paper, we obtained the strong convergence of Wong-Zakai approximations of reflected SDEs in a general multidimensional domain giving an affirmative answer to the question posed in [ES].
The strong convergence of Wong-Zakai approximations of the solution to the reflecting stochastic differential equations was studied in [2]. We continue the study and prove the strong convergence under weaker assumptions on the domain.
In this paper, we build the equivalence between rough differential equations driven by the lifted $G$-Brownian motion and the corresponding Stratonovich type SDE through the Wong-Zakai approximation. The quasi-surely convergence rate of Wong-Zakai approximation to $G-$SDEs with mesh-size $\frac{1}{n}$ in the $α$-Hölder norm is estimated as $(\frac{1}{n})^{\frac12-}.$ As corollary, we obtain the quasi-surely continuity of the above RDEs with respect to uniform norm.
This paper is devoted to the smooth and stationary Wong-Zakai approximations for a class of rough differential equations driven by a geometric fractional Brownian rough path $\boldsymbolω$ with Hurst index $H\in(\frac{1}{3},\frac{1}{2}]$. We first construct the approximation $\boldsymbolω_δ$ of $\boldsymbolω$ by probabilistic arguments, and then using the rough path theory to obtain the Wong-Zakai approximation for the solution on any finite interval. Finally, both the original system and approximative system generate a continuous random dynamical systems $φ$ and $φ^δ$. As a consequence of the…
We examine a Wong-Zakai type approximation of a family of stochastic differential equations driven by a general cadlag semimartingale. For such an approximation, compared with the pointwise convergence result by Kurtz, Pardoux and Protter [12, Theorem 6.5], we establish stronger convergence results under the Skorokhod M_1-topology, which, among other possible applications, implies the convergence of the first passage time of the solution to the stochastic differential equation.
In this work, we demonstrate the Wong-Zakai approximation results for two and three dimensional stochastic convective Brinkman-Forchheimer (SCBF) equations forced by Hilbert space valued Wiener noise on bounded domains. Even though the existence and uniqueness of a pathwise strong solution to SCBF equations is known, the existence of a unique solution to the approximating system is not immediate from the solvability results of SCBF equations, and we prove it by using Faedo-Galerkin approximation technique and monotonicity arguments. Moreover, as an application of the Wong-Zakai approximation, …
In this paper, we study the Wong-Zakai approximation of the solution to the stochastic differential equation on a domain $D$ in a Euclidean space with normal reflection at the boundary. We prove the $L^p$ convergence of the approximation in $C([0,T]\to \bar{D})$ under some general conditions on $D$.
The paper deals with convergence of solutions of a class of stochastic differential equations driven by infinite-dimensional semimartingales. The infinite-dimensional semimartingales considered in the paper are Hilbert-space valued. The theorems presented generalize the convergence result obtained by Wong and Zakai for stochastic differential equations driven by linear interpolations of a finite-dimensional Brownian motion. In particular, a general form of the correction factor is derived. Examples are given illustrating the use of the theorems to obtain other kinds of approximation results.
On Time Uniform Wong-Zakai Approximation Theorems
该研究针对随机微分方程Wong-Zakai逼近的均值误差随时域指数爆炸的问题，通过谱条件推导出具定量性质的时间一致收敛定理与误差界，填补了该类扩散逼近长时间一致误差分析的空白。
We construct Wong--Zakai approximations of time--inhomogeneous stochastic differential equations with regime switching (RSSDEs), and provide a convergence rate. %Given a family of finite-variation processes $\{\mathcal{F}^λ\}_{λ\ge 0}$ that converge strongly to a standard Brownian motion $\mathcal{B}$, we construct pathwise approximations for regime-switching, time-inhomogeneous stochastic differential equations in the Wong-Zakai sense. Moreover, we determine the rate of strong convergence to the solutions of such regime-switching SDEs, showing that this rate is almost as good as that of $\{\m…
In this article we study effects that small perturbations in the noise have to the solution of differential equations driven by Hölder continuous functions of order $H>\frac12$. As an application, we consider stochastic differential equations driven by a fractional Brownian motion. We introduce a Wong--Zakai type stationary approximation to the fractional Brownian motions and prove that it converges in a suitable space. Moreover, we provide sharp results on the rate of convergence in the $p$-norm. Our stationary approximation is suitable for all values of $H\in (0,1)$.
We deal with a class of abstract nonlinear stochastic models with multiplicative noise, which covers many 2D hydrodynamical models including the 2D Navier-Stokes equations, 2D MHD models and 2D magnetic Bénard problems as well as some shell models of turbulence. Our main result describes the support of the distribution of solutions. Both inclusions are proved by means of a general Wong-Zakai type result of convergence in probability for non linear stochastic PDEs driven by a Hilbert-valued Brownian motion and some adapted finite dimensional approximation of this process.
Wong--Zakai approximation of regime-switching SDEs via rough path theory
该研究借助粗糙路径理论，把 Wong–Zakai 逼近的收敛结果推广到 $\mathbb{R}^d$ 值机制转换随机微分方程，并保持 Nguyen 与 Peralta（2021）中的速率修正。
In this paper, we obtain the existence of random attractors for a class of evolution equations driven by a geometric fractional Brownian rough path with Hurst index $H\in(\frac{1}{3},\frac{1}{2}]$ and establish the upper semi-continuity of random attractors $\mathcal{A}_η$ for the approximated systems of the evolution equations.
We investigate the regularity of the law of Wong-Zakai-type approximations for Itô stochastic differential equations. These approximations solve random differential equations where the diffusion coefficient is Wick-multiplied by the smoothed white noise. Using a criteria based on the Malliavin calculus we establish absolute continuity and a Fokker-Planck-type equation solved in the distributional sense by the density. The parabolic smoothing effect typical of the solutions of Itô equations is lacking in this approximated framework; therefore, in order to prove absolute continuity, the initial …
We establish a unconditional and optimal strong convergence rate of Wong--Zakai type approximations in Banach space norm for a parabolic stochastic partial differential equation with monotone drift, including the stochastic Allen--Cahn equation, driven by an additive Brownian sheet. The key ingredient in the analysis is the fully use of additive nature of the noise and monotonicity of the drift to derive a priori estimation for the solution of this equation, in combination with the factorization method and stochastic calculus in martingale type 2 Banach spaces applied to deduce sharp error est…
In a recent article Lanconelli and Scorolli (2021) extended to the multidimensional case a Wong-Zakai-type approximation for Itô stochastic differential equations proposed by \Oksendal and Hu (1996). The aim of the current paper is to extend the latter result to system of stochastic differential equations of Itô type driven by fractional Brownian motion (fBm) like those considered by Hu (2018). The covariance structure of the fBm precludes us from using the same approach as that used by Lanconelli and Scorolli and instead we employ a truncated Cameron-Martin expansion as the approximation for …
We consider a class of stochastic differential equations driven by a one dimensional Brownian motion and we investigate the rate of convergence for Wong-Zakai-type approximated solutions. We first consider the Stratonovich case, obtained through the point-wise multiplication between the diffusion coefficient and a smoothed version of the noise; then, we consider Itô equations where the diffusion coefficient is Wick-multiplied by the regularized noise. We discover that in both cases the speed of convergence to the exact solution coincides with the speed of convergence of the smoothed noise towa…
We study stochastic differential equations (SDEs) with multiplicative Stratonovich-type noise of the form $ dX_t = b(X_t) dt + σ(X_t)\circ d W_t, X_0=x_0\in\mathbb{R}^d, t\geq0,$ with a possibly singular drift $b\in L^{p}(\mathbb{R}^d)$, $p>d$ and $p\geq 2$, and show that such SDEs can be approximated by random ordinary differential equations by smoothing the noise and the singular drift at the same time. We further prove a support theorem for this class of SDEs in a rather simple way using the Girsanov theorem.
Wong-Zakai approximation of density functions
本文结合粗糙路径理论与 Watanabe 分布 Malliavin 分析中的 Hu-Watanabe 定理，证明分数布朗粗糙路径驱动的 RDE 固定时刻密度函数的 Wong-Zakai 逼近，H=1/2 的经典情形亦获新结果。
A new numerical scheme for Itô stochastic differential equations based on Wick-type Wong-Zakai arguments
本文提出一种基于Wick型Wong-Zakai论证的无漂移一维Itô型随机微分方程新数值格式，通过多边形逼近平滑白噪声并对扩散系数一阶泰勒展开实现显式求解，具有一阶强收敛性且对线性扩散系数精确。
Wong-Zakai approximations and support theorems for SDEs under Lyapunov conditions
该研究在温和 Lyapunov 条件下证明 SDE 的 Stroock-Varadhan 型支撑定理，并通过新的 Wong-Zakai 逼近结果将适用范围扩展到多项式系数模型。
We prove a version of the Wong-Zakai theorem for one-dimensional parabolic nonlinear stochastic PDEs driven by space-time white noise. As a corollary, we obtain a detailed local description of solutions. Dedicated to the memory of Kiyosi Itô on the occasion of the 100th anniversary of his birth.
Wong-Zakai approximation for a class of SPDEs with fully local monotone coefficients and its application
本文针对具有完全局部单调系数的随机偏微分方程，通过Faedo-Galerkin方法等建立Wong-Zakai逼近，并应用于确定解的拓扑支集，涵盖多种流体模型。
We prove a Wong-Zakai theorem for the defocusing mass-critical stochastic nonlinear Schrödinger equation (NLS) on $\mathbb{R}$. The main ingredient are careful mixtures of bootstrapping arguments at both deterministic and stochastic levels. Several subtleties arising from the proof mark the difference between the dispersive case and corresponding situations in SDEs and parabolic stochastic PDEs, as well as the difference between the large-$n$ case and the limiting ($n=\infty$) case.
Wong-Zakai approximations with convergence rate for stochastic partial differential equations
本文研究了Wong-Zakai近似在随机偏微分方程中的收敛速率，并通过实例验证了其有效性。
In this paper we consider the following stochastic partial differential equation (SPDE) in the whole space: $du (t, x) = [a^{i j} (t, x) D_{i j} u(t, x) + f(u, t, x)]\, dt + \sum_{k = 1}^m g^k (u(t, x)) dw^k (t).$ We prove the convergence of a Wong-Zakai type approximation scheme of the above equation in the space $ C^{θ} ([0, T], H^γ_p (\mathbb{R}^d)) $ in probability, for some $ θ\in (0,1/2), γ\in (1, 2)$, and $p > 2$. We also prove a Stroock-Varadhan's type support theorem. To prove the results we combine V. Mackevicius ideas from his papers on Wong-Zakai theorem and the support theorem for…
We discuss a class of stochastic second-order PDEs in one space-dimension with an inner boundary moving according to a possibly non-linear, Stefan-type condition. We show that proper separation of phases is attained, i.e., the solution remains negative on one side and positive on the other side of the moving interface, when started with the appropriate initial conditions. To extend results from deterministic settings to the stochastic case, we establish a Wong-Zakai type approximation. After a coordinate transformation the problems are reformulated and analysed in terms of stochastic evolution…
Strong uniform Wong--Zakai approximations of Lévy-driven Marcus SDEs
本文研究了Lévy驱动的$d$维Marcus随机微分方程的Wong--Zakai型逼近格式，证明了其具有$\frac12$阶强收敛性，并确定了局部一致强收敛的速率为$h^{\frac{1-\varepsilon}{4d}}$。
Upon its inception the theory of regularity structures allowed for the treatment for many semilinear perturbations of the stochastic heat equation driven by space-time white noise. When the driving noise is non-Gaussian the machinery of theory can still be used but must be combined with an infinite number of stochastic estimates in order to compensate for the loss of hypercontractivity. In this paper we obtain a more streamlined and automatic set of criteria implying these estimates which facilitates the treatment of some other problems including non-Gaussian noise such as some general phase c…
Dynamics and Wong-Zakai approximations of stochastic nonlocal PDEs with long time memory
本文结合 Galerkin 方法与 Dafermos 变换研究带长时间记忆的加性噪声随机非局部 PDE，证明解的适定性与随机吸引子存在性，并建立 Wong-Zakai 逼近的收敛及吸引子上半连续性。
We extend to the multidimensional case a Wong-Zakai-type theorem proved by Hu and Øksendal in [7] for scalar quasi-linear Itô stochastic differential equations (SDEs). More precisely, with the aim of approximating the solution of a quasilinear system of Itô's SDEs, we consider for any finite partition of the time interval $[0,T]$ a system of differential equations, where the multidimensional Brownian motion is replaced by its polygonal approximation and the product between diffusion coefficients and smoothed white noise is interpreted as a Wick product. We remark that in the one dimensional ca…
Rate of convergence of Wong-Zakai approximations for stochastic partial differential equations
本文证明了由Wiener过程驱动的随机偏微分方程的Wong-Zakai逼近收敛速率本质上与逼近Wiener过程的驱动过程W_n的收敛速率相同，前提是W_n的面积过程也以该速率收敛到W的面积过程，并考虑了非退化及退化情形。
We study a family of numerical schemes applied to a class of multiscale systems of stochastic differential equations. When the time scale separation parameter vanishes, a well-known homogenization or Wong--Zakai diffusion approximation result states that the slow component of the considered system converges to the solution of a stochastic differential equation driven by a real-valued Wiener process, with Stratonovich interpretation of the noise. We propose and analyse schemes for effective approximation of the slow component. Such schemes satisfy an asymptotic preserving property and generaliz…
Wong--Zakai Convergence and FNO Learning for Stochastic Volterra Equations
该研究针对描述带记忆随机系统的随机Volterra积分方程，建立Wong--Zakai收敛性，将镜像填充傅里叶神经算子的解算子学习框架扩展至该类方程，经数值实验验证其预测精度与泛化能力。
Lévy areas, Wong Zakai anomalies in diffusive limits of Deterministic Lagrangian Multi-Time Dynamics
本文研究流体拉格朗日动力学扩散极限中噪声解释导致的稳定性差异，比较 Itô、Stratonovich 与 Wong–Zakai 极限及 Lévy 面积离散化，并借助 SALT 均匀化推导解释异常漂移的物理来源。
Averaging Dynamics and Wong-Zakai approximations for a Fast-Slow Navier-Stokes System Driven by fractional Brownian Motion
本文研究了由分数布朗运动驱动的快慢纳维-斯托克斯系统，通过粗糙路径理论分析其极限行为，揭示了不同H值下慢分量收敛到不同类型的纳维-斯托克斯系统的特性。
Consider the heat equation driven by a smooth, Gaussian random potential: \begin{align*} \partial_t u_{\varepsilon}=\tfrac12Δu_{\varepsilon}+u_{\varepsilon}(ξ_{\varepsilon}-c_{\varepsilon}), \ \ t>0, x\in\mathbb{R}, \end{align*} where $ξ_{\varepsilon}$ converges to a spacetime white noise, and $c_{\varepsilon} $ is a diverging constant chosen properly. For any $ n\geq 1 $, we prove that $ u_{\varepsilon} $ converges in $ L^n $ to the solution of the stochastic heat equation. Our proof is probabilistic, hence provides another perspective of the general result of Hairer and Pardoux \cite{Hairer1…
In this paper we prove a general approximation result for reflected stochastic differential equations in bounded domains satisfying conditions reorganized by Ren and Wu. Then we show that it includes Wong-Zakai approximation, mollifier approximation, etc.
We study the problem of existence, uniqueness and approximation of solutions of finite dimensional Stratonovich stochastic differential equations with reflecting boundary condition driven by semimartingales with jumps. As an application we generalize known results on the Wong-Zakai type approximations.
We propose a novel small time approximation for the solution to the Zakai equation from nonlinear filtering theory. We prove that the unnormalized filtering density is well described over short time intervals by the solution of a deterministic partial differential equation of Kolmogorov type; the observation process appears in a pathwise manner through the degenerate component of the Kolmogorov's type operator. The rate of convergence of the approximation is of order one in the lenght of the interval. Our approach combines ideas from Wong-Zakai-type results and Wiener chaos approximations for …
We study stochastic model reduction for evolution equations in infinite dimensional Hilbert spaces, and show the convergence to the reduced equations via abstract results of Wong-Zakai type for stochastic equations driven by a scaled Ornstein-Uhlenbeck process. Both weak and strong convergence are investigated, depending on the presence of quadratic interactions between reduced variables and driving noise. Finally, we are able to apply our results to a class of equations used in climate modeling.
We try to justify rigorously, using a Wong-Zakai approximation argument, the susceptible-infected-susceptible (SIS) stochastic differential equation proposed in [2]. We discover that according to this approach the "right" stochastic model to be considered should be the Stratonovich version of the Itô equation analyzed in [2]. Surprisingly, this alternative model presents the following feature: the threshold value characterizing the two different asymptotic regimes of the solution coincides with the one describing the classical SIS deterministic equation.
We present a Feynman-Kac formula for the $1$-dimensional stochastic heat equation (SHE) driven by a time-homogeneous Gaussian white noise potential, where the noise is interpreted in the Wick-Itô-Skorokhod sense. Our approach consists in constructing a Wong-Zakai-type approximation for the SHE from which we are able to obtain an "approximating Feynman-Kac" representation via the reduction of the approximated SHE to a deterministic partial differential equation (PDE). Then we will show that those "approximating Feynman-Kac" converge to a well defined object we will call "formal Feynman-Kac" rep…
We consider anticipative Stratonovich stochastic differential equations driven by some stochastic process (not necessarily a semi-martingale). No adaptedness of initial point or vector fields is assumed. Under a simple condition on the stochastic process, we show that the unique solution of the above SDE understood in the rough path sense is actually a Stratonovich solution. This condition is satisfied by the Brownian motion and the fractional Brownian motion with Hurst parameter greater than 1/4. As application, we obtain rather flexible results such as support theorems, large deviation princ…
We consider the stochastic transport linear equation and we prove existence and uniqueness of weak $L^{p}-$solutions. Moreover, we obtain a representation of the general solution and a Wong-Zakai principle for this equation. We make only minimal assumptions, similar to the deterministic problem. The proof is supported on the generalized Itô-Ventzel-Kunita formula (see 15) and the theory of Lions-DiPerna on transport linear equation (see 9).
We give a Wong-Zakai type characterisation of the solutions of quasilinear heat equations driven by space-time white noise in $1+1$ dimensions. In order to show that the renormalisation counterterms are local in the solution, a careful arrangement of a few hundred terms is required. The main tool in this computation is a general integration by parts formula that provides a number of linear identities for the renormalisation constants.
We study a nonlinear stochastic partial differential equation whose solution is the conditional log-Laplace functional of a superprocess in a random environment. We establish its existence and uniqueness by smoothing out the nonlinear term and making use of the particle system representation developed by Kurtz and Xiong [Stochastic Process. Appl. 83 (1999) 103-126]. We also derive the Wong-Zakai type approximation for this equation. As an application, we give a direct proof of the moment formulas of Skoulakis and Adler [Ann. Appl. Probab. 11 (2001) 488-543].
We prove that stochastic replicator dynamics can be interpreted as intrinsic Brownian motion on the simplex equipped the Aitchison geometry. As an immediate consequence we derive three approximation results in the spirit of Wong-Zakai approximation, Donsker's invariance principle and a JKO-scheme. Finally, using the Fokker-Planck equation and Wasserstein-contraction estimates, we study the long time behavior of the stochastic replicator equation, as an example of a non-gradient drift diffusion on the Aitchison simplex.
Fundamental weak convergence theorem for stochastic Volterra integral equations and its applications
针对非马尔可夫随机Volterra积分方程，结合马尔可夫提升、泰勒展开和Fr\'echet微分，建立基本弱收敛定理，统一分析数值方法的弱误差，首次获得Wong-Zakai近似的弱收敛阶。
In this paper we prove an approximate continuity result for stochastic differential equations with normal reflections in domains satisfying Saisho's conditions, which together with the Wong-Zakai approximation result completes the support theorem for such diffusions in the uniform convergence topology. Also by adapting Millet and Sanz-Solé's idea, we characterize in Hölder norm the support of diffusions reflected in domains satisfying the Lions-Sznitman conditions by proving limit theorems of adapted interpolations. Finally we apply the support theorem to establish a boundary-interior maximum …
The long time behavior of Wong-Zakai approximations of 2D as well as 3D non-autonomous stochastic convective Brinkman-Forchheimer (CBF) equations with non-linear diffusion terms on bounded and unbounded ($\mathbb{R}^d$ for $d=2,3$) domains is discussed in this work. To establish the existence of random pullback attractors, the concept of asymptotic compactness (AC) is used. In bounded domains, AC is proved via compact Sobolev embeddings. In unbounded domains, due to the lack of compact embeddings, the ideas of energy equations and uniform tail estimates are exploited to prove AC. In the litera…
Under the key assumption of finite ρ-variation, ρ\in[1,2), of the covariance of the underlying Gaussian process, sharp a.s. convergence rates for approximations of Gaussian rough paths are established. When applied to Brownian resp. fractional Brownian motion (fBM), ρ=1 resp. ρ=1/(2H), we recover and extend the respective results of [Hu--Nualart; Rough path analysis via fractional calculus; TAMS 361 (2009) 2689-2718] and [Deya--Neuenkirch--Tindel; A Milstein-type scheme without Lévy area terms for SDEs driven by fractional Brownian motion; AIHP (2011)]. In particular, we establish an a.s. rate…
We consider anticipative Stratonovich stochastic differential equations driven by some stochastic process lifted to a rough path. Neither adaptedness of initial point and vector fields nor commuting conditions between vector field is assumed. Under a simple condition on the stochastic process, we show that the unique solution of the above SDE understood in the rough path sense is actually a Stratonovich solution. We then show that this condition is satisfied by the Brownian motion. As application, we obtain rather flexible results such as support theorems, large deviation principles and Wong--…
In this paper, we study the stochastic Hamiltonian flow in Wasserstein manifold, the probability density space equipped with $L^2$-Wasserstein metric tensor, via the Wong--Zakai approximation. We begin our investigation by showing that the stochastic Euler-Lagrange equation, regardless it is deduced from either variational principle or particle dynamics, can be interpreted as the stochastic kinetic Hamiltonian flows in Wasserstein manifold. We further propose a novel variational formulation to derive more general stochastic Wassersetin Hamiltonian flows, and demonstrate that this new formulati…
Similar to ordinary differential equations, rough paths and rough differential equations can be formulated in a Banach space setting. For $α\in (1/3,1/2)$, we give criteria for when we can approximate Banach space-valued weakly geometric $α$-rough paths by signatures of curves of bounded variation, given some tuning of the Hölder parameter. We show that these criteria are satisfied for weakly geometric rough paths on Hilbert spaces. As an application, we obtain Wong-Zakai type result for function space valued martingales using the notion of (unbounded) rough drivers.
We establish finite time extinction with probability one for weak solutions of the Cauchy-Dirichlet problem for the 1D stochastic porous medium equation with Stratonovich transport noise and compactly supported smooth initial datum. Heuristically, this is expected to hold because Brownian motion has average spread rate $O(t^\frac{1}{2})$ whereas the support of solutions to the deterministic PME grows only with rate $O(t^{\frac{1}{m{+}1}})$. The rigorous proof relies on a contraction principle up to time-dependent shift for Wong-Zakai type approximations, the transformation to a deterministic P…
The Wong-Zakai theorem asserts that ODEs driven by "reasonable" (e.g. piecewise linear) approximations of Brownian motion converge to the corresponding Stratonovich stochastic differential equation. With the aid of rough path analysis, we study "non-reasonable" approximations and go beyond a well-known criterion of [Ikeda--Watanabe, North Holland 1989] in the sense that our result applies to perturbations on all levels, exhibiting additional drift terms involving any iterated Lie brackets of the driving vector fields. In particular, this applies to the approximations by McShane ('72) and Sussm…
In this paper we consider the Stratonovich reflected stochastic differential equation $dX_t=σ(X_t)\circ dW_t+b(X_t)dt+dL_t$ in a bounded domain $Ø$ which satisfies conditions, introduced by Lions and Sznitman, which are specified below. Letting $W^N_t$ be the $N$-dyadic piecewise linear interpolation of $W_t$ what we show is that one can solve the reflected ordinary differential equation $\dot X^N_t=σ(X^N_t)\dot W^N_t+b(X^N_t)+\dot L^N_t$ and that the distribution of the pair $(X^N_t,L^N_t)$ converges weakly to that of $(X_t,L_t)$. Hence, what we prove is a distributional version for reflected…
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