Peter K. Friz
Lyons' Rough Path theory is currently formulated in p-variation topology. We extend his main-result, the Universal Limit Theorem, to a stronger Hoelder topology. Several approximations to Brownian Rough Paths are studied. As application of our approach, we obtain the celebrated support theorem (in Hoelder-topology) as immediate (!) corollary.
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We formulate indefinite integration with respect to an irregular function as an algebraic problem and provide a criterion for the existence and uniqueness of a solution. This allows us to define a good notion of integral with respect to irregular paths with Hoelder exponent greater than 1/3 (e.g. samples of Brownian motion) and study the problem of the existence, uniqueness and continuity of solution of differential equations driven by such paths. We recover Young's theory of integration and the main results of Lyons' theory of rough paths in Hoelder topology.
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.
This article shows an Itô-Wentzell type formula adapted to rough paths with $α$-Hölder regularity $α\in (\frac{1}{3},\frac{1}{2}]$. We improve previous results of R. Castrequini and P. Catuogno for the Young integral and C. Keller and J. Zhang for rough paths.
Rough Functional Itô Formula
本文证明了适用于$α$-Hölder连续路径的路径依赖泛函的粗糙Itô公式，核心方法结合了sewing引理与路径依赖导数的Taylor逼近。
Based on a dyadic approximation of Itô integrals, we show the existence of Itô càdlàg rough paths above general semimartingales, suitable Gaussian processes and non-negative typical price paths. Furthermore, Lyons-Victoir extension theorem for càdlàg paths is presented, stating that every càdlàg path of finite $p$-variation can be lifted to a rough path.
We use simple sub-Riemannian techniques to prove that an arbitrary geometric p-rough path in the sense of Lyons (98) is the limit in sup-norm of a sequence of canonically lifted smooth paths, which are uniformly bounded in p-variation, clarifying the two different definitions of a geometric p-rough path, Lyons (98), Lyons/Qian (02). Our proofs are based on fine estimates in terms of control functions and are sufficiently general to include the case of Hoelder- and modulus-type regularity, Friz/Victoir (03). This allows us to extend a few classical results on Hoelder-spaces (Ciesielski, Musiela…
Itô formula for reduced rough paths
本文在粗糙路径理论框架下推广约化粗糙路径的 Itô 公式，将适用粗糙度范围扩展至更奇异的 $\frac{1}{4}<α\leq\frac{1}{3}$。
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。
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 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.
Recently, Hairer--Pillai proposed the notion of $θ$-roughness of a path which leads to a deterministic Norris lemma. In the Gubinelli framework (Hoelder, level 2) of rough paths, they were then able to prove a Hoermander type result (SDEs driven by fractional Brownian motion, $H>1/3$). We take a step back and propose a natural "roughness" condition relative to a given $p$-rough path in the sense of Lyons; the aim being a Doob-Meyer result for rough integrals in the sense of Lyons. The interest in our (weaker) condition is that it is immediately verified for large classes of Gaussian processes,…
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…
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…
Randomisation of rough stochastic differential equations
本文研究粗糙随机微分方程（RSDEs）的随机化，统一分析粗糙Ito过程在随机化后转化为“双重随机”Ito过程的条件、机制及其条件分布。
Motivated by a problematic coming from mathematical finance, this paper is devoted to existing and additional results of continuity and differentiability of the Itô map associated to rough differential equations. These regularity results together with Malliavin calculus are applied to sensitivities analysis for stochastic differential equations driven by multidimensional Gaussian processes with continuous paths, especially fractional Brownian motions. Precisely, in that framework, results on computation of greeks for Itô's stochastic differential equations are extended. An application in mathe…
Jump Itô-type formula with arbitrary regularity
本文提出一种适用于具有有限p-变差且存在跳跃的路径的伊藤型公式，区分了粗糙积分与跳跃修正项，并在连续和 càdlàg 情况下分别推导了相应公式，同时给出了随机微分方程的链式法则和路径上的对数财富分解。
Unified Optimality Conditions for Stochastic Optimal Control in the Rough Path and Itô Frameworks
本文建立了Itô与粗糙路径框架下随机最优控制PMP条件的统一形式，通过条件期望桥接伴随方程，并应用于生成模型微调与反馈控制。
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.
Functional Itô-formula and Taylor expansions for non-anticipative maps of càdlàg rough paths
本文利用 càdlàg 粗糙路径签名逼近性质建立泛函 Itô 公式，并通过 Marcus 变换处理跳跃扰动，进而得到非预期映射的泛函 Taylor 展开，推广了 Dupire 等人的近期结果。
A geometric p-rough path can be seen to be a genuine path of finite p-variation with values in a Lie group equipped with a natural distance. The group and its distance lift (R^{d},+,0) and its Euclidean distance. This approach allows us to easily get a precise modulus of continuity for the Enhanced Brownian Motion (the Brownian Motion and its Levy Area). As a first application, extending an idea due to Millet & Sanz-Sole, we characterize the support of the Enhanced Brownian Motion (without relying on correlation inequalities). Secondly, we prove Schilder's theorem for this Enhanced Brownian Mo…
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。
We show in this note that the Ito-Lyons solution map associated to a rough differential equation is Frechet differentiable when understood as a map between some Banach spaces of controlled paths. This regularity result provides an elementary approach to Taylor-like expansions of Inahama-Kawabi type for solutions of rough differential equations depending on a small parameter, and makes the construction of some natural dynamics on the path space over any compact Riemannian manifold straightforward, giving back Driver's flow as a particular case.
Whenever an Itô-Wentsel type of formula holds for composition of flows of a certain differential dynamics, there exists locally a decomposition of the corresponding flow according to complementary distributions (or foliations, in the case of integrability of these distributions). Many examples have been proved in distinct context of dynamics: Stratonovich stochastic equations, Lévy driven noise, low regularity $α$-Hölder control functions ($ α\in (1/2,1]$), see e.g. [6], [7], [20], [21]. Here we present the proof of this categorical property: we illustrate with the $α$-Hölder rough path, $α\in…
We investigate existence, uniqueness and regularity for solutions of rough parabolic equations of the form $\partial _tu-A_tu-f=(\dot X_t(x) \cdot \nabla + \dot Y_t(x))u$ on $[0,T]\times\mathbb{R}^d.$ To do so, we introduce a concept of "differential rough driver", which comes with a counterpart of the usual controlled paths spaces in rough paths theory, built on the Sobolev spaces $W^{k,p}.$ We also define a natural notion of geometricity in this context, and show how it relates to a product formula for controlled paths. In the case of transport noise (i.e.\ when $Y=0$), we use this framework…
Rough stochastic differential equations
本文提出将伊藤随机理论与洛伊恩粗糙微分理论相结合，以解决路径随机滤波、控制及随机系统分析中的共同噪声问题。
T. Lyons' rough path theory is something like a deterministic version of K. Ito's theory of stochastic differential equations, combined with ideas from K. T. Chen's theory of iterated path integrals. In this article we survey rough path theory, in particular, its probabilistic aspects.
Malliavin Calculus is about Sobolev-type regularity of functionals on Wiener space, the main example being the Ito map obtained by solving stochastic differential equations. Rough path analysis is about strong regularity of solution to (possibly stochastic) differential equations. We combine arguments of both theories and discuss existence of a density for solutions to stochastic differential equations driven by a general class of non-degenerate Gaussian processes, including processes with sample path regularity worse than Brownian motion.
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…
Pathwise central limit theorem and moderate deviations via rough paths for SPDEs with multiplicative noise
本研究借助粗糙路径理论提出通用框架，推导小乘性线性噪声扰动的随机偏微分方程的路径中心极限定理与中偏差原理，给出最优收敛速度，并应用于多类典型随机偏微分方程。
Universal approximation with signatures of non-geometric rough paths
本文提出通过扩展路径与二次变分项的签名，实现非几何粗糙路径的通用逼近，并应用于金融模型校准与期权定价。
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…
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…
Fractional Sobolev spaces, also known as Besov or Slobodetzki spaces, arise in many areas of analysis, stochastic analysis in particular. We prove an embedding into certain q-variation spaces and discuss a few applications. First we show q-variation regularity of Cameron-Martin paths associated to fractional Brownian motion and other Volterra processes. This is useful, for instance, to establish large deviations for enhanced fractional Brownian motion. Second, the q-variation embedding, combined with results of rough path theory, provides a different route to a regularity result for stochastic…
Branched Itô formula and natural Itô-Stratonovich isomorphism
本文提出分支伊藤公式与自然伊藤-斯特拉托诺维奇同构，扩展了半鞅的修正公式，通过分支粗糙路径理论建立代数结构特征化。
Gamma Hedging without Rough Paths
本文提出无需粗路径理论即可理解Gamma对冲的鲁棒性，并通过分区序列的p次变分和泰勒定理推导出经典Delta对冲结果，同时应用于障碍期权和亚洲期权。
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 establish a new scale of $p$-variation estimates for martingale paraproducts, martingale transforms, and Itô integrals, of relevance in rough paths theory, stochastic, and harmonic analysis. As an application, we introduce rough semimartingales, a common generalization of classical semimartingales and (controlled) rough paths, and their integration theory.
This work focuses on the Laplace approximation for the rough differential equation (RDE) driven by mixed rough path with as . Firstly, based on geometric rough path lifted from mixed fractional Brownian motion (fBm), the Schilder-type large deviation principle (LDP) for the law of the first level path of the solution to the RDE is given. Due to the particularity of mixed rough path, the main difficulty in carrying out the Laplace approximation is to prove the Hilbert-Schmidt property for the Hessian matrix of the Itô map restricted on the Cameron-Martin space of the mixed fBm. To this end, we …
Large deviation principle for slow-fast rough differential equations via controlled rough paths
本文在受控粗糙路径框架下，针对由混合分数布朗运动驱动的慢-快粗糙微分方程证明了大偏差原理，通过变分表示将问题转化为受控系统的定性性质，并利用快方程不变测度平均与解映射连续性证明受控慢分量的弱收敛。
A rough path approach to pathwise stochastic integration à la Föllmer
本文用粗糙路径扩展 Föllmer 路径积分框架，在二次变差与 Lévy area 存在时统一刻画 Itô、Stratonovich 和后向 Itô 积分，并给出分割不变性的充要条件。
In this article, we consider limit theorems for some weighted type random sums (or discrete rough integrals). We introduce a general transfer principle from limit theorems for unweighted sums to limit theorems for weighted sums via rough path techniques. As a by-product, we provide a natural explanation of the various new asymptotic behaviors in contrast with the classical unweighted random sum case. We apply our principle to derive some weighted type Breuer-Major theorems, which generalize previous results to random sums that do not have to be in a finite sum of chaos. In this context, a Breu…
A mixed fractional CIR model: positivity and an implicit Euler scheme
本文提出了一种混合分数布朗运动驱动的CIR模型，证明了其在Feller条件下严格正性，并建立了隐式欧拉方案的收敛性。
We consider additive functionals of stationary Markov processes and show that under Kipnis-Varadhan type conditions they converge in rough path topology to a Stratonovich Brownian motion, with a correction to the Levy area that can be described in terms of the asymmetry (non-reversibility) of the underlying Markov process. We apply this abstract result to three model problems: First we study random walks with random conductances under the annealed law. If we consider the Ito rough path, then we see a correction to the iterated integrals even though the underlying Markov process is reversible. …
A Direct Approach to Rough Paths in Time-Weighted Besov Spaces
本文扩展Besov粗糙路径框架至时间加权空间，建立加权缝补引理并构造积分，证明Itô-Lyons映射的Lipschitz连续性，实现位置敏感的误差控制，并以奇异系数扩散为例。
We present two different approaches to stochastic integration in frictionless model free financial mathematics. The first one is in the spirit of Itô's integral and based on a certain topology which is induced by the outer measure corresponding to the minimal superhedging price. The second one is based on the controlled rough path integral. We prove that every "typical price path" has a naturally associated Itô rough path, and justify the application of the controlled rough path integral in finance by showing that it is the limit of non-anticipating Riemann sums, a new result in itself. Compar…
Consider a multidimensional diffusion process $X=\{X\left(t\right) :t\in\lbrack0,1]\}$. Let $\varepsilon>0$ be a \textit{deterministic}, user defined, tolerance error parameter. Under standard regularity conditions on the drift and diffusion coefficients of $X$, we construct a probability space, supporting both $X$ and an explicit, piecewise constant, fully simulatable process $X_{\varepsilon}$ such that \[ \sup_{0\leq t\leq1}\left\Vert X_{\varepsilon}\left(t\right) -X\left(t\right) \right\Vert_{\infty}<\varepsilon \] with probability one. Moreover, the user can adaptively choose $\varepsilon^…
We consider uniformly subelliptic operators on certain unimodular Lie groups of polynomial growth. It was shown by Saloff-Coste and Stroock that classical results of De Giorgi, Nash, Moser, Aronson extend to this setting. It was then observed by Sturm that many proofs extend naturally to the setting of locally compact Dirichlet spaces. We relate these results to what is known as rough path theory by showing that they provide a natural and powerful analytic machinery for construction and study of (random) geometric Hoelder rough paths. (In particular, we obtain a simple construction of the Lyon…
We introduce a canonical method for transforming a discrete sequential data set into an associated rough path made up of lead-lag increments. In particular, by sampling a $d$-dimensional continuous semimartingale $X:[0,1] \rightarrow \mathbb{R}^d$ at a set of times $D=(t_i)$, we construct a piecewise linear, axis-directed process $X^D: [0,1] \rightarrow\mathbb{R}^{2d}$ comprised of a past and future component. We call such an object the Hoff process associated with the discrete data $\{X_{t}\}_{t_i\in D}$. The Hoff process can be lifted to its natural rough path enhancement and we consider the…
We consider a system of differential equations in a fast long range dependent random environment and prove a homogenization theorem involving multiple scaling constants. The effective dynamics solves a rough differential equation, which is `equivalent' to a stochastic equation driven by mixed Itô integrals and Young integrals with respect to Wiener processes and Hermite processes. Lacking other tools we use the rough path theory for proving the convergence, our main technical endeavour is on obtaining an enhanced scaling limit theorem for path integrals (Functional CLT and non-CLT's) in a stro…
Universal approximation on non-geometric rough paths and applications to financial derivatives pricing
本文提出了一种基于多项式的粗糙路径逼近方法，用于金融衍生品定价中的无套利条件分析。
We develop a Fourier approach to rough path integration, based on the series decomposition of continuous functions in terms of Schauder functions. Our approach is rather elementary, the main ingredient being a simple commutator estimate, and it leads to recursive algorithms for the calculation of pathwise stochastic integrals, both of Itô and of Stratonovich type. We apply it to solve stochastic differential equations in a pathwise manner.
We prove an enhanced limit theorem for additive functionals of a multi-dimensional Volterra process $(y_t)_{t\geq 0}$ in the rough path topology. As an application, we establish weak convergence as $\varepsilon\to 0$ of the solution of the random ordinary differential equation (ODE) $\frac{d}{dt}x^\varepsilon_t=\frac{1}{\sqrt \varepsilon} f(x_t^\varepsilon,y_{\frac{t}{\varepsilon}})$ and show that its limit solves a rough differential equation driven by a Gaussian field with a drift coming from the Lévy area correction of the limiting rough driver. Furthermore, we prove that the stochastic flo…
Consider a mean-reverting equation, generalized in the sense it is driven by a 1-dimensional centered Gaussian process with Hölder continuous paths on $[0,T]$ ($T > 0$). Taking that equation in rough paths sense only gives local existence of the solution because the non-explosion condition is not satisfied in general. Under natural assumptions, by using specific methods, we show the global existence and uniqueness of the solution, its integrability, the continuity and differentiability of the associated Itô map, and we provide an $L^p$-converging approximation with a rate of convergence ($p\ge…
We consider the stochastic continuity equation associated to an Itô diffusion with irregular drift and diffusion coefficients. We give regularity conditions under which weak solutions are renormalized in the sense of DiPerna/Lions, and prove well-posedness in $L^p$. As an application, we give a new proof of renormalizability (hence uniqueness) of weak solutions to the stochastic continuity equation when the diffusion matrix is constant and the drift only belongs to $L^q_tL^p$, where $\frac{2}{q} + \frac{n}{p} <1$, without resorting to the regularity of the stochastic flow or a duality method.
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…
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。
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…
A Càdlàg Rough Path Foundation for Robust Finance
本文基于粗糙路径理论为随机Itô积分提供路径wise基础，引入càdlàg路径的Property (RIE)以证明粗糙积分存在性，并验证多种交易策略和价格路径的适用性。
The geometry of controlled rough paths
本文证明任意阶受控（分支）粗糙路径空间构成连续 Banach 空间场，建立内蕴拓扑与逼近结果，并使粗糙积分及 Itô-Lyons 映射连续且保持结构。
Unbiased Rough Integrators and No Free Lunch in Rough-Path-Based Market Models
本文研究了粗糙路径理论在支持无摩擦连续无免费午餐市场中的能力，推导了无偏粗糙积分器的分类，并指出该框架受限于经典半鞅范式。
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