Anna Ananova
We establish the existence of solutions to path-dependent rough differential equations with non-anticipative coefficients. Regularity assumptions on the coefficients are formulated in terms of horizontal and vertical derivatives.
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In this article we consider rough differential equations (RDEs) driven by non-geometric rough paths, using the concept of branched rough paths introduced in Gubinelli (2004). We first show that branched rough paths can equivalently be defined as $γ$-Hölder continuous paths in some Lie group, akin to geometric rough paths. We then show that every branched rough path can be encoded in a geometric rough path. More precisely, for every branched rough path $\mathbf{X}$ lying above a path $X$, there exists a geometric rough path $\bar{\mathbf{X}}$ lying above an extended path $\bar X$, such that $\b…
Itô formula for planarly branched rough paths
本文针对包含更多随机过程的平面分支粗糙路径，证明了粗糙度$\frac{1}{4}< α\leq \frac{1}{2}$条件下的Itô公式，扩展了粗糙路径理论。
We introduce a notion of rough paths on embedded submanifolds and demonstrate that this class of rough paths is natural. On the way we develop a notion of rough integration and an efficient and intrinsic theory of rough differential equations (RDEs) on manifolds. The theory of RDEs is then used to construct parallel translation along manifold valued rough paths. Finally, this framework is used to show there is a one to one correspondence between rough paths on a d-dimensional manifold and rough paths on d-dimensional Euclidean space. This last result is a rough path analogue of Cartan's develo…
Based on two isomorphisms of Hopf algebras, we provide a bound in the optimal order on the remainder of the truncated Taylor expansion for controlled differential equations driven by branched rough paths.
Rough differential equations and planarly branched universal limit theorem
该研究针对粗糙路径理论中通用极限定理的适用范围问题，利用巴拿赫不动点定理，证明了粗糙度在$\frac{1}{4}<α≤\frac{1}{3}$区间的平面分支粗糙路径的通用极限定理，拓展了该定理的适用场景。
We provide a necessary and sufficient condition for a rough control driving a differential equation to be reconstructable, to some order, from observing the resulting controlled evolution. Physical examples and applications in stochastic filtering and statistics demonstrate the practical relevance of our result.
The Lipschitz continuity of the solution to branched rough differential equations
本文利用Grossman Larson Hopf代数与张量Hopf代数的同构，将次黎曼几何技术用于分支粗糙微分方程，证明了解对初值、向量场和驱动粗糙路径的显式Lipschitz连续性。
We show in this note how the machinery of C^1-approximate flows devised in the work "Flows driven by rough paths", and applied there to reprove and extend most of the results on Banach space-valued rough differential equations driven by a finite dimensional rough path, can be used to deal with rough differential equations driven by an infinite dimensional Banach space-valued weak geometric Holder p-rough paths, for any p>2, giving back Lyons' theory in its full force in a simple way.
We show in this work how the machinery of C^1-approximate flows introduced in our previous work "Flows driven by rough paths", provides a very efficient tool for proving well-posedness results for path-dependent rough differential equations on flows of the form dϕ= V h(dt) + F X(dt), for smooth enough path-dependent vector fields V,F = (V_1,...,V_\ell), any Holder weak geometric p-rough path X and any a-Holder path h, with a+1/p>1.
These are lecture notes for a Master 2 course on rough differential equations driven by weak geometric Holder p-rough paths, for any p>2. They provide a short, self-contained and pedagogical account of the theory, with an emphasis on flows. The theory is illustrated by some now classical applications to stochastic analysis, such as the basics of Freidlin-Wentzel theory of large deviations for diffusions, or Stroock and Varadhan support theorem.
Rough differential equations in the flow approach
本文提出利用树索引的流方法，证明分数布朗运动驱动的粗随机微分方程在Hurst指数H > 1/4时的适定性。
We address propagation of chaos for large systems of rough differential equations associated with random rough differential equations of mean field type $$ dX_t = V(X_t,\mathcal{L}(X_t))dt + F(X_t,\mathcal{L}(X_t))dW_t $$ where $W$ is a random rough path and $\mathcal{L}(X_t)$ is the law of $X_t$. We prove propagation of chaos, and provide also an explicit optimal convergence rate. The analysis is based upon the tools we developed in our companion paper [1] for solving mean field rough differential equations and in particular upon a corresponding version of the Itô-Lyons continuity theorem. Th…
A theory of differential equations driven by a non-differentiable path has recently been developed by Lyons. We develop an alternative approach to this theory, using (modified Euler approximations), and investigate its applicability to stochastic differential equations driven by Brownian motion. We also give some other examples showing that the main results are reasonably sharp.
Flows driven by multi-indices Rough Paths
本文提出了一种由多索引粗糙路径驱动的粗糙微分方程的解理论，并利用log-ODE方法进行分析。
The non-linear sewing lemma constructs flows of rough differential equations from a braod class of approximations called almost flows. We consider a class of almost flows that could be approximated by solutions of ordinary differential equations, in the spirit of the backward error analysis. Mixing algebra and analysis, a Taylor formula with remainder and a composition formula are central in the expansion analysis. With a suitable algebraic structure on the non-smooth vector fields to be integrated, we recover in a single framework several results regarding high-order expansions for various ki…
Recently it was proved that the group of rough paths modulo tree-like equivalence is isomorphic to the corresponding signature group through the signature map S (a generalized notion of taking iterated path integrals). However, the proof of this uniqueness result does not contain any information on how to "see" the trajectory of a (tree-reduced) rough path from its signature, and a constructive understanding on the uniqueness result (in particular on the inverse of S) has become an interesting and important question. The aim of the present paper is to reconstruct a rough path from its signatur…
Functional differential equations driven by càdlàg rough paths
本文针对由 càdlàg 粗糙路径驱动的路径依赖粗糙微分方程，建立了解的存在唯一性及 Itô-Lyons 映射的局部 Lipschitz 连续性，并将结果应用于受控、带延迟的 RDEs 等，还构造了对应 Itô 积分的联合粗糙路径。
We prove some results, which are used in arXiv:1406.7871, about weakly geometric rough paths that are well-known in finite dimensions, but need proof in the infinite dimensional setting.
Rough differential equations and reduced rough paths: a Lie bracket characterization
研究 Hölder 正则性\(\frac13<\alpha\le\frac12\)下粗糙微分方程，从约化粗糙路径角度出发，确定其确定解的障碍，证明在交换向量场情形下反对称 Lévy 面积无关，此为研究粗糙微分方程提供新视角。
We extend the recently developed rough path theory for Volterra equations from (Harang and Tindel, 2019) to the case of more rough noise and/or more singular Volterra kernels. It was already observed in (Harang and Tindel, 2019) that the Volterra rough path introduced there did not satisfy any geometric relation, similar to that observed in classical rough path theory. Thus, an extension of the theory to more irregular driving signals requires a deeper understanding of the specific algebraic structure arising in the Volterra rough path. Inspired by the elements of "non-geometric rough paths" d…
We exhibit an explicit natural isomorphism between spaces of branched and geometric rough paths. This provides a multi-level generalisation of the isomorphism of Lejay-Victoir (2006) as well as a canonical version of the Itô-Stratonovich correction formula of Hairer-Kelly (2015). Our construction is elementary and uses the property that the Grossman-Larson algebra is isomorphic to a tensor algebra. We apply this isomorphism to study signatures of branched rough paths. Namely, we show that the signature of a branched rough path is trivial if and only if the path is tree-like, and construct a no…
We consider controlled differential equations and give new estimates for higher order Euler schemes. Our proofs are inspired by recent work of A. M. Davie who considers first and second order schemes. In order to implement the general case we make systematic use of geodesic approximations in the free nilpotent group. As application, we can control moments of solutions to rough path differential equations (RDEs) driven by random rough paths with sufficient integrability and have a criteria for L^q - convergence in the Universal Limit Theorem. We also obtain Azencott type estimates and asymptoti…
We provide in this work a robust solution theory for random rough differential equations of mean field type $$ dX_t = V(X_t,\mathcal{L}(X_t))dt + F(X_t,\mathcal{L}(X_t))dW_t, $$ where $W$ is a random rough path and $\mathcal{L}(X_t)$ stands for the law of $X_t$, with mean field interaction in both the drift and diffusivity. The analysis requires the introduction of a new rough path-like setting and an associated notion of controlled path. We use crucially Lions' approach to differential calculus on Wasserstein space along the way.
Rough path theory
本文综述粗糙路径理论，该理论为低正则项微分方程提供连续性框架，并在随机分析中为不规则路径驱动的随机微分方程赋予新意义，涵盖解的概念、路径签名及应用。
In this paper, we establish the theory of nonlinear rough paths. We give the definition of nonlinear rough paths, and develop the integrals. Then, we study differential equations driven by nonlinear rough paths. Afterwards, we compare the nonlinear rough paths and classic theory of linear rough paths. Finally, we apply this theory to rough partial differential equations.
A partial rough path space for rough volatility
本文为粗糙波动率模型构建偏粗糙路径空间，证明路径级大偏差原理，描述短期隐含波动率极限行为。
We develop the rough path counterpart of Itô stochastic integration and - differential equations driven by general semimartingales. This significantly enlarges the classes of (Itô / forward) stochastic differential equations treatable with pathwise methods. A number of applications are discussed.
In this paper, we study reflected differential equations driven by continuous paths with finite $p$-variation ($1\le p<2$) and $p$-rough paths ($2\le p<3$) on domains in Euclidean spaces whose boundaries may not be smooth. We define reflected rough differential equations and prove the existence of a solution. Also we discuss the relation between the solution to reflected stochastic differential equation and reflected rough differential equation when the driving process is a Brownian motion.
Rough path theory and an introduction to rough partial differential equations
本文介绍粗偏微分方程的基本理论，涵盖粗路径理论及其在随机偏微分方程中的应用。
We prove existence of global solutions for differential equations driven by a geometric rough path under the condition that the vector fields have linear growth. We show by an explicit counter-example that the linear growth condition is not sufficient if the driving rough path is not geometric. This settle a long-standing open question in the theory of rough paths. So in the geometric setting we recover the usual sufficient condition for differential equation. The proof rely on a simple mapping of the differential equation from the Euclidean space to a manifold to obtain a rough differential e…
We prove existence and uniqueness of the solution of a one-dimensional rough differential equation driven by a step-2 rough path and reflected at zero. In order to deal with the lack of control of the reflection measure the proof uses some ideas we introduced in a previous work dealing with rough kinetic PDEs [arXiv:1604.00437].
The purpose of this article is to solve rough differential equations with the theory of regularity structures. These new tools recently developed by Martin Hairer for solving semi-linear partial differential stochastic equations were inspired by the rough path theory. We take a pedagogical approach to facilitate the understanding of this new theory. We recover results of the rough path theory with the regularity structure framework. Hence, we show how to formulate a fixed point problem in the abstract space of modelled distributions to solve the rough differential equations. We also give a pro…
We investigate rough differential equations with a time-dependent reflecting lower barrier, where both the driving (rough) path and the barrier itself may have jumps. Assuming the driving signals allow for Young integration, we provide existence, uniqueness and stability results. When the driving signal is a càdlàg $p$-rough path for $p \in [2,3)$, we establish existence to general reflected rough differential equations, as well as uniqueness in the one-dimensional case.
This paper introduces the path derivatives, in the spirit of Dupire's functional Itô calculus, for the controlled paths in the rough path theory with possibly non-geometric rough paths. The theory allows us to deal with rough integration and rough PDEs in the same manner as standard stochastic calculus. We next study rough PDEs with coefficients depending on the rough path itself, which corresponds to stochastic PDEs with random coefficients. Such coefficients is less regular in the time variable and is not covered in the existing literature. The results are useful for studying viscosity solut…
Integration of branched rough paths
本文研究了分支p-粗糙路径积分的构造及其在粗糙路径度量下的连续性，并证明了其与相关Π-粗糙路径积分的一致性。
We construct a procedure for Bogoliubov-Parasiuk-Hepp-Zimmermann (BPHZ) renormalization of a rough path in view of the relation between rough path theory and regularity structure. We also provide a plain expression of the BPHZ-renormalized model in a rough path. BPHZ renormalization plays a central role in the theory of singular stochastic partial differential equations and assures the convergence of the model in the regularity structure. Here we demonstrate that the renormalization is also effective in a rough path setting by considering its application to rough path theory and mathematical f…
In line with the notion of probabilistic rough paths introduced in the previous contribution \cite{salkeld2021Probabilistic}, we address corresponding random controlled rough paths (first introduced in \cite{2019arXiv180205882.2B}), the structure of which is indexed by Lions forests. These are statistical distributions over the space of paths described by the combination of a jet on the underlying probabilistic rough path and a remainder term. The regularity of the latter facilitates the definition of a rough integral. We establish closedness and stability of two key operators on random contro…
This article introduces the splitting method to systems responding to rough paths as external stimuli. The focus is on nonlinear partial differential equations with rough noise but we also cover rough differential equations. Applications to stochastic partial differential equations arising in control theory and nonlinear filtering are given.
Optimal extension to Sobolev rough paths
该研究证明满足α>1/p>0的Sobolev路径可提升为Lyons意义下的Sobolev粗糙路径，且存在关于严格凸泛函的唯一最优提升，给出了最小范数提升与布朗运动Stratonovich提升的实例，并提及Besov空间推广。
In this work we study rough differential equations driven by a fractional Brownian motion with Hurst parameter H>1/4 and establish Varadhan's small time estimates for the density of solutions of such equations under Hormander's type conditions.
Smooth rough paths, their geometry and algebraic renormalization
本文引入光滑粗糙路径，利用Maurer-Cartan视角实现Lyons延拓定理的纯代数形式及粗糙路径重整化，并扩展至拟几何与Hopf代数框架。
We introduce the space of rough paths with Sobolev regularity and the corresponding concept of controlled Sobolev paths. Based on these notions, we study rough path integration and rough differential equations. As main result, we prove that the solution map associated to differential equations driven by rough paths is a locally Lipschitz continuous map on the Sobolev rough path space for any arbitrary low regularity $α$ and integrability $p$ provided $α>1/p$.
Approximate Transitivity of Young Translation on Rough Paths
本文证明在粗糙路径空间中Young平移具有稠密轨道，并应用于随机级数和Gaussian过程（包括非中心分数布朗粗糙路径）的全支撑准则。
We analyze common lifts of stochastic processes to rough paths/rough drivers-valued processes and give sufficient conditions for the cocycle property to hold for these lifts. We show that random rough differential equations driven by such lifts induce random dynamical systems. In particular, our results imply that rough differential equations driven by the lift of fractional Brownian motion in the sense of Friz-Victoir induce random dynamical systems.
In this paper we study rough differential equations driven by Gaussian rough paths from the viewpoint of Malliavin calculus. Under mild assumptions on coefficient vector fields and underlying Gaussian processes, we prove that solutions at a fixed time is smooth in the sense of Malliavin calculus. Examples of Gaussian processes include fractional Brownian motion with Hurst parameter larger than $1/4$.
Moderate deviations for rough differential equations
本文研究了由Hurst参数在1/4至1/2之间的缩放分数布朗粗糙路径驱动的粗糙微分方程，证明了当缩放参数趋于零时该方程的中偏差原理。
Regularization by noise for rough differential equations driven by Gaussian rough paths
本文研究了由Gaussian粗糙路径驱动的粗糙微分方程，在非确定性和一致椭圆性条件下证明了低正则性漂移的逐路径适定性，并利用流变换和Malliavin计算确定了分数布朗运动下漂移的Hölder连续性界限。
We propose a quantitative direct method of proving the stability result for Gaussian rough differential equations in the sense of Gubinelli \cite{gubinelli}. Under the strongly dissipative assumption of the drift coefficient function, we prove that the trivial solution of the system under small noise is exponentially stable.
We propose a quantitative direct method of proving the stability result for Gaussian rough differential equations in the sense of Gubinelli \cite{gubinelli}. Under the strongly dissipative assumption of the drift coefficient function, we prove that the trivial solution of the system under small noise is exponentially stable.
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.
We prove an extension to the classical continuity theorem in rough paths. We show that two $p$-rough paths are close in all levels of iterated integrals provided the first $\lfl p \rfl$ terms are close in a uniform sense. Applications include estimation of the difference of the signatures of two uniformly close paths and convergence rates of Gaussian rough paths.
In this paper, we build the foundation for a theory of controlled rough paths on manifolds. A number of natural candidates for the definition of manifold valued controlled rough paths are developed and shown to be equivalent. The theory of controlled rough one-forms along such a controlled path and their resulting integrals are then defined. This general integration theory does require the introduction of an additional geometric structure on the manifold which we refer to as a "parallelism." The transformation properties of the theory under change of parallelisms is explored. Using these trans…
We consider rough paths with jumps. In particular, the analogue of Lyons' extension theorem and rough integration are established in a jump setting, offering a pathwise view on stochastic integration against cadlag processes. A class of Levy rough paths is introduced and characterized by a sub-ellipticity condition on the left-invariant diffusion vector fields and and a certain integrability property of the Carnot--Caratheodory norm with respect to the Levy measure on the group, using Hunt's framework of Lie group valued Levy processes. Examples of Levy rough paths include standard multi-dimen…
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.
In this article we investigate the rough paths structure of a process $X_t$ living in a fixed Wiener chaos. Specifically, we formulate various types of rough lifts of $X_t$ and study their properties. As application, we study the integrabilities of quantities related to rough differential equations driven by $X_t$.
We study the Small Ball Probabilities (SBPs) of Gaussian rough paths. While many works on rough paths study the Large Deviations Principles (LDPs) for stochastic processes driven by Gaussian rough paths, it is a noticeable gap in the literature that SBPs have not been extended to the rough path framework. LDPs provide macroscopic information about a measure for establishing Integrability type properties. SBPs provide microscopic information and are used to establish a locally accurate approximation for a measure. Given the compactness of a Reproducing Kernel Hilbert space (RKHS) ball, its metr…
In this paper we prove the derivative process of a rough differential equation driven by Brownian rough path has finite $L^r$-moment for any $r /ge 1$. Thanks to Burkholder-Davis-Gundy's inequality, this kind of problem is easy in the usual SDE theory. In the context of rough path theory, however, it does not seem so obvious.
Reflected backward stochastic differential equations with rough drivers
本文研究带粗糙驱动项的反射倒向随机微分方程，证明其适定性、惩罚逼近性质，并将其应用于带障碍粗糙PDE的粘性解理论及粗糙路径框架下的美式期权定价与最优停止问题。
Random dynamical systems generated by rough differential equations driven by controlled rough paths
本文研究由受控粗糙路径驱动的粗糙微分方程，通过典范提升建立路径wise适定性，构造可微解余圈，并证明局部稳定、不稳定和中心流形的不变性。
Backward stochastic differential equations (BSDEs) in the sense of Pardoux-Peng [Backward stochastic differential equations and quasilinear parabolic partial differential equations, Lecture Notes in Control and Inform. Sci., 176, 200--217, 1992] provide a non-Markovian extension to certain classes of non-linear partial differential equations; the non-linearity is expressed in the so-called driver of the BSDE. Our aim is to deal with drivers which have very little regularity in time. To this end we establish continuity of BSDE solutions with respect to rough path metrics in the sense of Lyons […
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