Yves LeJan, Zhongmin Qian
The signature of Brownian motion in $\mathbb{R}^{d}$ over a running time interval $[0,T]$ is the collection of all iterated Stratonovich path integrals along the Brownian motion. We show that, in dimension $d\geq 2$, almost all Brownian motion sample paths (running up to time $T$) are determined by its signature over $[0,T]$
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Let $B$ be a fractional Brownian motion with Hurst parameter $H=1/6$. It is known that the symmetric Stratonovich-style Riemann sums for $\int g(B(s))\,dB(s)$ do not, in general, converge in probability. We show, however, that they do converge in law in the Skorohod space of càdlàg functions. Moreover, we show that the resulting stochastic integral satisfies a change of variable formula with a correction term that is an ordinary Itô integral with respect to a Brownian motion that is independent of $B$.
In the present paper, we prove that with probability one, the Stratonovich signatures of a multidimensional diffusion process (possibly degenerate) over [0,1], which is the collection of all iterated Stratonovich's integrals of the diffusion process over [0,1], determine the diffusion sample paths.
We show how to build an immersion coupling of a two-dimensional Brownian motion $(W_1, W_2)$ along with $\binom{n}{2} + n= \tfrac12n(n+1)$ integrals of the form $\int W_1^iW_2^j \circ dW_2$, where $j=1,\ldots,n$ and $i=0, \ldots, n-j$ for some fixed $n$. The resulting construction is applied to the study of couplings of certain hypoelliptic diffusions (driven by two-dimensional Brownian motion using polynomial vector fields). This work follows up previous studies concerning coupling of Brownian stochastic areas and time integrals (Ben Arous, Cranston and Kendall (1995), Kendall and Price (2004…
Time-dependency in hyperbolic Anderson model: Stratonovich regime
本文研究时间依赖高斯噪声驱动的双曲安德森方程的可解性与大t渐近性，通过Stratonovich矩界、时间随机化布朗相交局部时表示及大偏差原理三项进展，得到解存在的充要条件与解期望的大t极限形式。
Rough Path Renormalization from Stratonovich to Itô for Fractional Brownian Motion
本文基于 Lyons 粗糙路径框架，为 Hurst 参数在 (1/3, 1/2] 区间的分数布朗运动建立 Itô 型路径积分理论，并将其应用于分数 Black-Scholes 模型和高维分数 Ornstein-Uhlenbeck 过程，解决了非半鞅过程的积分难题。
The isometry of symmetric-Stratonovich integrals w.r.t. Fractional Brownian motion $H< \frac{1}{2}$
本文针对参数 1/4<H<1/2 的多维分数布朗运动驱动的对称-Stratonovich 积分，在不施加 Malliavin 正则性条件下完整刻画了实现 L^2 等距的被积过程 Hilbert 空间，并给出 Gubinelli 粗糙积分 L^2 范数的显式表达式。
In this paper, we study rough path properties of stochastic integrals of Itô's type and Stratonovich's type with respect to $G$-Brownian motion. The roughness of $G$-Brownian Motion is estimated and then the pathwise Norris lemma in $G$-framework is obtained.
Given a Gaussian process $X$, its canonical geometric rough path lift $\mathbf{X}$, and a solution $Y$ to the rough differential equation (RDE) $\mathrm{d}Y_{t} = V\left (Y_{t}\right ) \circ \mathrm{d} \mathbf{X}_t$, we present a closed-form correction formula for $\int Y \circ \mathrm{d} \mathbf{X} - \int Y \, \mathrm{d} X$, i.e. the difference between the rough and Skorohod integrals of $Y$ with respect to $X$. When $X$ is standard Brownian motion, we recover the classical Stratonovich-to-It{ô} conversion formula, which we generalize to Gaussian rough paths with finite $p$-variation, $p < 3$…
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…
Hyperbolic Anderson equations with general time-independent Gaussian noise: Stratonovich regime
本文研究时不变高斯噪声驱动的双曲型Anderson方程，证明Dalang条件为解存在的充要条件，并在齐次性假设下建立解的长时间与高阶矩精确渐近，方法基于Laplace变换下Stratonovich矩的表示与大偏差技巧。
The purpose of this paper is to establish the convergence in law of the sequence of "midpoint" Riemann sums for a stochastic process of the form f'(W), where W is a Gaussian process whose covariance function satisfies some technical conditions. As a consequence we derive a change-of-variable formula in law with a second order correction term which is an Itô integral of f''(W) with respect to a Gaussian martingale independent of W. The proof of the convergence in law is based on the techniques of Malliavin calculus and uses a central limit theorem for q-fold Skorohod integrals, which is a multi…
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.
The signature of a $d$-dimensional Brownian motion is a sequence of iterated Stratonovich integrals along the Brownian paths, an object taking values in the tensor algebra over $\RR^{d}$. In this note, we derive the exact rate of convergence for the expected signatures of piecewise linear approximations to Brownian motion. The computation is based on the identification of the set of words whose coefficients are of the leading order, and the convergence is concentrated on this subset of words. Moreover, under the choice of projective tensor norm, we give the explicit value of the leading term c…
We develop the functional Itô/path-dependent calculus with respect to fractional Brownian motion with Hurst parameter $H> \frac{1}{2}$. Firstly, two types of integrals are studied. The first type is Stratonovich integral, and the second type is Wick-Itô integral. Then we establish the functional Itô formulas for fractional Brownian motion, which extend the functional Itô formulas in Dupire (2009) and Cont-Fournié (2013) to the case of non-semimartingale. Finally, as an application, we deal with a class of fractional backward stochastic differential equations (BSDEs). A relation between fractio…
In this paper we study the Stratonovich stochastic differential equation $\mathrm{d} X=|X|^α\circ\mathrm{d} B$, $α\in(-1,1)$, which has been introduced by Cherstvy et al. [New Journal of Physics 15:083039 (2013)] in the context of analysis of anomalous diffusions in heterogeneous media. We determine its weak and strong solutions, which are homogeneous strong Markov processes \chng{spending zero time at $0$: for $α\in (0,1)$, these solutions have the form $$ X_t^θ=\bigl((1-α)B_t^θ\bigr)^{1/(1-α)}, $$ where $B^θ$ is the $θ$-skew Brownian motion driven by $B$ and starting at $\frac{1}{1-α}(X_0)^{…
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…
Based on the recent development of the framework of Volterra rough paths, we consider here the probabilistic construction of the Volterra rough path associated to the fractional Brownian motion with $H>\frac{1}{2}$ and for the standard Brownian motion. The Volterra kernel $k(t,s)$ is allowed to be singular, and behaving similar to $|t-s|^{-γ}$ for some $γ\geq 0$. The construction is done in both the Stratonovich and Itô sense. It is based on a modified Garsia-Rodemich-Romsey lemma which has an interest in its own right, as well as tools from Malliavin calculus. A discussion of challenges and p…
Stochastic selection problem for a Stratonovich SDE with power non-linearity
本文研究幂非线性Stratonovich随机微分方程在零处多解的选择问题，证明添加小的外部加性噪声可恢复唯一性，并在噪声趋于零时得到物理对称情形下的非均匀扩散。
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--…
Given a solution $Y$ to a rough differential equation (RDE), a recent result [8] extends the classical Itö-Stratonovich formula and provides a closed-form expression for $\int Y \circ \mathrm{d} \mathbf{X} - \int Y \, \mathrm{d} X$, i.e. the difference between the rough and Skorohod integrals of $Y$ with respect to $X$, where $X$ is a Gaussian process with finite $p$-variation less than 3. In this paper, we extend this result to Gaussian processes with finite $p$-variation such that $3 \leq p < 4$. The constraint this time is that we restrict ourselves to Volterra Gaussian processes with kerne…
In this article we establish a large deviation principle for the family {ν_ε:ε\in (0,1)} of distributions of the scaled stochastic processes {P_{-\log\sqrtε}Z_t}_{t\leq 1}, where (Z_t)_{t\in \lbrack 0,1]} is a square-integrable martingale over Brownian filtration and (P_t)_{t\geq 0} is the Ornstein-Uhlenbeck semigroup. The rate function is identified as well in terms of the Wiener-Itô chaos decomposition of the terminal value Z_{1}. The result is established by developing a continuity theorem for large deviations, together with two essential tools, the hypercontractivity of the Ornstein-Uhlenb…
Weak approximation for Gaussian processes from renewal processes
本文研究了从再生成过程构建高斯过程近似的方法，证明了某些进程可用于近似分数布朗运动等，并提供了确保收敛的充分条件。
We study the relationship between mixed stochastic differential equations and the corresponding rough path equations driven by standard Brownian motion and fractional Brownian motion with Hurst parameter $H>1/2$. We establish a correction formula, which relates both types of equations, analogously to the Itō-Stratonovich correction formula. This correction formula allows to transfer properties, which are established for one type of equation to the other, and we will illustrate this by considering numerical methods for mixed and rough SDEs
In this paper, we study different scaling rough path limit regimes in space and time for the Langevin dynamics on a quasi-planar fluctuating Helfrich surfaces. The convergence results of the processes were already proven in the work by Duncan, Elliott, Pavliotis and Stuart (2015). We extend this work by proving the convergence of the Itô and Stratonovich rough path lift. For the rough path limit, there appears, typically, an area correction term to the Itô iterated integrals, and in certain regimes to the Stratonovich iterated integrals. This yields additional information on the homogenization…
We quantize a multidimensional $SDE$ (in the Stratonovich sense) by solving the related system of $ODE$'s in which the $d$-dimensional Brownian motion has been replaced by the components of functional stationary quantizers. We make a connection with rough path theory to show that the solutions of the quantized solutions of the $ODE$ converge toward the solution of the $SDE$. On our way to this result we provide convergence rates of optimal quantizations toward the Brownian motion for $\frac 1q$-H\" older distance, $q>2$, in $L^p(¶)$.
Approximation of Random Differential Equations Driven by Physical Brownian Motion with Fast Oscillating Noise
研究由满足奇异摄动朗之万方程且具有缩放混合随机力的半鞅驱动的随机微分方程逼近。用扩散逼近法探索半鞅粗糙路径提升的极限，应用通用极限定理确定随机微分方程极限，还建立了混合随机力的迭代弱不变原理。
The signature of a path γis a sequence whose n-th term is the order-n iterated integrals of γ. It arises from solving multidimensional linear differential equations driven by γ. We are interested in relating the path properties of γwith its signature. If γis C^{1}, then an elegant formula of Hambly and Lyons relates the length of γto the tail asymptotics of the signature. We show an analogous formula for the multidimensional Brownian motion, with the quadratic variation playing a similar role to the length. In the proof, we study the hyperbolic development of Brownian motion and also obtain a …
The solution of rough differential equation, driven by the Itô signature of a continuous local martingale, exists uniquely a.s. when the vector field is Lip(β) for β> 1, and coincides a.s. with the Itô signature of the solution of parallel stochastic differential equation. Moreover, the Itô solution can be recovered pathwisely by concatenating discounted Stratonovich solutions.
A representation for the Expected Signature of Brownian motion up to the first exit time of the planar unit disc
针对平面单位圆盘首离时刻前的布朗运动期望签名问题，受其部分项的贝塞尔函数显式公式启发，通过研究Lyons-Ni偏微分方程推导张量级数表示，简化后得到各张量分量多项式首阶项公式。
Affine Structure of the Brownian Signature
本文通过张量代数上的Riccati方程，建立了时间增广布朗签名的无限维仿射变换理论，揭示了其广义仿射结构，并提出了随机化Riccati方程以恢复全局表示。
On Itô-Stratonovich formula for rough sheets
提出简化平面粗糙微积分的新策略，通过基本泰勒展开减少迭代积分，在非对称 Hölder 正则性下推广经典平面变量变换公式。
A Quasi-sure Non-degeneracy Property for the Brownian Rough Path
本文研究粗糙路径理论中的签名唯一性相关问题，证明拟必然意义下布朗运动签名路径不自交，进而推出布朗粗糙路径无树状片段、样本路径可由签名在重参数化下唯一确定的结论。
On the geometric Brownian motion with state-dependent variable exponent diffusion term
该研究提出含状态依赖可变指数 $p(\cdot)$ 的随机模型以统一推广 GBM 与 CEV，证明解的存在唯一性，并给出其相对 GBM 的逐路径误差上界及解析、数值验证。
For every $d\geq 1$, we consider the $d$-dimensional Hermitian fractional Brownian motion (HfBm), that is the process with values in the space of $(d\times d)$-Hermitian matrices and with upper-diagonal entries given by complex fractional Brownian motions of Hurst index $H\in (0,1)$. We follow the approach of [A. Deya and R. Schott: On the rough paths approach to non-commutative stochastic calculus, JFA (2013)] to define a natural integral with respect to the HfBm when $H>\frac13$, and identify this interpretation with the rough integral with respect to the $d^2$ entries of the matrix. Using t…
In this work we present different results concerning the signature and the cubature of fractional Brownian motion (fBm). The first result regards the rate of convergence of the expected signature of the linear piecewise approximation of the fBm to its exact value, for a value of the Hurst parameter $H\in(\frac{1}{2},1)$. We show that the rate of convergence is given by $2H$. We believe that this rate is sharp as it is consistent with the result of Ni and Xu, who showed that the sharp rate of convergence for the Brownian motion (i.e. fBm with $H=\frac{1}{2}$) is given by $1$. The second result …
The signature of a path provides a top down description of the path in terms of its effects as a control [Differential Equations Driven by Rough Paths (2007) Springer]. The signature transforms a path into a group-like element in the tensor algebra and is an essential object in rough path theory. The expected signature of a stochastic process plays a similar role to that played by the characteristic function of a random variable. In [Chevyrev (2013)], it is proved that under certain boundedness conditions, the expected value of a random signature already determines the law of this random signa…
Expected Signature on a Riemannian Manifold and Its Geometric Implications
本文在紧致黎曼流形上利用布朗桥与布朗环的期望签名渐近重构距离函数与曲率，并推导签名动力学的内在偏微分方程。
We pursue our investigations, initiated in [8], about stochastic integration with respect to the non-commutative fractional Brownian motion (NC-fBm). Our main objective in this paper is to compare the pathwise constructions of [8] with a Skorohod-type interpretation of the integral. As a first step, we provide details on the basic tools and properties associated with non-commutative Malliavin calculus, by mimicking the presentation of Nualart's celebrated treatise [14]. Then we check that, just as in the classical (commutative) situation, Skorohod integration can indeed be considered in the pr…
The expected signature is an analogue of the Laplace transform for rough paths. Chevyrev and Lyons showed that, under certain moment conditions, the expected signature determines the laws of signatures. Lyons and Ni posed the question of whether the expected signature of Brownian motion up to the exit time of a domain satisfies Chevyrev and Lyons' moment condition. We provide the first example where the answer is negative.
The aim of this paper is to study the $d$-dimensional stochastic heat equation with a multiplicative Gaussian noise which is white in space and it has the covariance of a fractional Brownian motion with Hurst parameter $% H\in (0,1)$ in time. Two types of equations are considered. First we consider the equation in the Itô-Skorohod sense, and later in the Stratonovich sense. An explicit chaos development for the solution is obtained. On the other hand, the moments of the solution are expressed in terms of the exponential moments of some weighted intersection local time of the Brownian motion.
Optimal sample path properties of stochastic processes often involve generalized Hölder- or variation norms. Following a classical result of Taylor, the exact variation of Brownian motion is measured in terms of $ψ(x) \equiv $ $x^{2}/\log \log (1/x) $ near $0+$. Such $ψ$-variation results extend to classes of processes with values in abstract metric spaces. (No Gaussian or Markovian properties are assumed.) To establish integrability properties of the $ψ$-variation we turn to a large class of Gaussian rough paths (e.g. Brownian motion and Lévy's area viewed as a process in a Lie group) and pro…
Tail Asymptotics of the Signature of various stochastic processes and its connection to the Quadratic Variation
本文扩展了布朗运动签名与二次变差联系的研究，证明了半鞅Itô签名与流逝时间及分数布朗运动fWIS签名与Hurst参数的联系，并提出了多维扩展猜想。
Path-dependent processes from signatures
本文利用时间增广布朗运动的路径签名，为随机线性 Volterra 和延迟方程等路径依赖积分方程提供了显式级数展开，揭示了其无限维马尔可夫结构并提出了简单的数值近似方案。
General Signature Kernels
本文提出通用签名核，阐释其可解释为PDE解的平均值并可通过求积规则计算，推导了含布朗运动期望Stratonovich签名的闭式公式，建立其与双曲展开映射的新关联，并评估了其在维纳测度非参数拟合优度检验中的应用。
The goal of this paper is to simplify and strengthen the Le Jan-Qian approximation scheme of studying the uniqueness of signature problem to the non-Markov setting. We establish a general framework for a class of multidimensional stochastic processes over [0,1] under which with probability one, the signature (the collection of iterated path integrals in the sense of rough paths) is well-defined and determines the sample paths of the process up to reparametrization. In particular, by using the Malliavin calculus we show that our method applies to a class of Gaussian processes including fraction…
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…
Geometric Interpretation of Brownian Motion on Riemannian Manifolds
本文提出黎曼流形上布朗运动的统一几何框架，通过沿正交标架注噪并设计匹配Laplace–Beltrami算子的漂移项构建随机微分方程，推导两种随机积分表述并明确漂移的几何对应，为非线性空间扩散过程提供自洽基础。
It is proved that the mean signature of multi-dimensional fractional brownian motion admits a meromorphic continuation in the hurst parameter to the entire complex plane. Each contstituent mean iterated integral is a sum of hypergeometric integrals indexed by the pair partitions which refine the partition arising from the sequential list of integrands which defines it. Furthermore, each such hypergeometric integral is holomorphic in the complement of a finite union of rational progressions determined by the combinatorial structure of the pair partition which defines it. It is not proved that t…
On the Wiener Chaos Expansion of the Signature of a Gaussian Process
该研究推导了含H∈(1/4,1)的分数布朗运动在内的一类高斯过程签名的维纳混沌分解，得到其期望签名表达式，将已有相关公式推广到H>1/4的一般情形，解决了公开问题，还研究了连续中心化高斯半鞅。
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…
Global universal approximation with Brownian signatures
研究在合适粗糙路径空间上一般路径依赖和非预期泛函的\(L^p\)通用逼近定理，通过推导加权粗糙路径空间的全局通用逼近定理，证明其适用于高斯过程，可逼近布朗滤波适应的\(p\)可积随机过程。
Rough analysis of two scale systems
本文利用粗糙路径理论分析受加性 Brown 噪声扰动的三维慢–快耦合 Navier–Stokes 系统，证明慢分量依分布收敛至含 Itô–Stokes 漂移与粗糙传输噪声的极限系统。
Infinite ergodicity for geometric Brownian motion
本文研究几何布朗运动及其推广的渐近分布特性，通过无限遍历性方法探讨其在不同离散化参数下的概率分布行为。
We study the density of the signature of fractional Brownian motions with parameter $H>1/4$. In particular, we prove existence, smoothness, global Gaussian upper bounds and Varadhan's type asymptotics for this density. A key result is that the estimates on the density we obtain are controlled by the Carnot-Carathéodory distance of the group.
Approximating the signature of Brownian motion for high order SDE simulation
本文研究带时间的布朗运动签名的近似方法，针对其非高斯、难模拟的非线性积分提出新方案并介绍近期方法，且展示了其在SDE高阶数值模拟中的应用。
Truncated Signature Information for Mixed Fractional Brownian Paths
研究Hurst指数大于1/4的混合分数布朗运动路径的有限期望签名信息，证明尺度权衡与分离及局部逆界。
Small mass limit of expected signature for physical Brownian motion
本文研究了物理布朗运动在小质量极限下的预期签名收敛问题，揭示了其在张量代数中的非平凡收敛特性，并通过SDE模型分析了相关数学结构。
Concentration of Truncated Signatures of Gaussian Rough Paths
本文研究高斯粗糙路径截断签名的集中性，揭示了截断级别与尾部重量的权衡，并推导了布朗运动和分数布朗运动的方差公式。
Expected signature of stopped Brownian motion on $d$-dimensional $C^{2, α}$-domains has finite radius of convergence everywhere: $2\leq d \leq 8$
本文研究了在二维至八维C^{2,α}域中停止布朗运动的期望签名收敛半径问题，证明其处处有限，并引入域平均双曲发展方法进行对称化。
The $L^q$-bounds for Derivatives of Unitary Developments of Random Continuous Geometric Rough Paths
本文推导随机连续几何粗糙路径酉发展导数的显式公式并建立$L^q$界，定量研究特征函数解析性，证明在温和条件下签名分布由期望签名决定，部分回答了Lyons和Ni的开放问题。
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