Shigeki Aida
We prove weak Poincare inequalities on domains which are inverse images of open sets in Wiener spaces under continuous functions of Brownian rough paths. The result is applicable to Dirichlet forms on loop groups and connected open subsets of path spaces over compact Riemannian manifolds.
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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 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.
Starting from the construction of a geometric rough path associated with a fractional Brownian motion with Hurst parameter $H\in]{1/4}, {1/2}[$ given by Coutin and Qian (2002), we prove a large deviation principle in the space of geometric rough paths, extending classical results on Gaussian processes. As a by-product, geometric rough paths associated to elements of the reproducing kernel Hilbert space of the fractional Brownian motion are obtained and an explicit integral representation is given.
This note extends the results in [8] to construct the rough super-Brownian motion on finite volume with Dirichlet boundary conditions. The backbone of this study is the convergence of discrete approximations of the parabolic Anderson model (PAM) on a box.
The present paper is devoted to the study of sample paths of G-Brownian motion and stochastic differential equations (SDEs) driven by G-Brownian motion from the view of rough path theory. As the starting point, we show that quasi-surely, sample paths of G-Brownian motion can be enhanced to the second level in a canonical way so that they become geometric rough paths of roughness 2 < p < 3. This result enables us to introduce the notion of rough differential equations (RDEs) driven by G-Brownian motion in the pathwise sense under the general framework of rough paths. Next we establish the funda…
In this work we study the smoothing effect of rough differential equations driven by a fractional Brownian motion with parameter $H>1/4$. The regularization estimates we obtain generalize to the fractional Brownian motion previous results by Kusuoka and Stroock and can be seen as a quantitative version of the existence of smooth densities under Hörmander's type conditions.
We consider the stochastic continuity equation perturbed by a fractional Brownian motion and the drift is allowed to be discontinuous. We show that for almost all paths of the fractional Brownian motion there exists a solution to the equation.
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…
We consider a geometric rough path associated with a fractional Brownian motion with Hurst parameter $H\in]{1/4}, {1/2}[$. We give an approximation result in a modulus type distance, up to the second order, by means of a sequence of rough paths lying above elements of the reproducing kernel Hilbert space.
We introduce the (path-valued) Brownian frame process whose evaluation at time t is the sample path of the underlying Brownian motion run from time t-1 to t. Due to its connections with Gaussian Volterra processes and SDDEs this is an interesting object to study. The first part deals with path-wise properties of the Brownian frame process in the p-variation norm. The second part shows the non-existence of a Levy area random variable in a particular norm, revealing the difficulty in establishing a Rough Path integration theory for the Brownian Frame process.
We survey existing results concerning the study in small times of the density of the solution of a rough differential equation driven by fractional Brownian motions. We also slightly improve existing results and discuss some possible applications to mathematical finance.
We construct in this article a rough path over fractional Brownian motion with arbitrary Hurst index by (i) using the Fourier normal ordering algorithm introduced in \cite{Unt-Holder} to reduce the problem to that of regularizing tree iterated integrals and (ii) applying the Bogolioubov-Parasiuk-Hepp-Zimmermann (BPHZ) renormalization algorithm to Feynman diagrams representing tree iterated integrals.
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。
We consider a rough differential equation indexed by a small parameter $\varepsilon>0$. When the rough differential equation is driven by fractional Brownian motion with Hurst parameter $H$ ($1/4<H<1/2$), we prove the Laplace-type asymptotics for the solution as the parameter $\varepsilon$ tends to zero.
A note on the continuity in the Hurst index of the solution of rough differential equations driven by a fractional Brownian motion
本文在粗糙路径框架下，证明了当Hurst参数H在(1/3, 1/2]区间内时，由分数布朗运动驱动的随机微分方程的解关于该参数具有连续性。
The Compact Support Property of Rough Super Brownian Motion on $\mathbb{R}^2$
本文研究了静态随机环境中分支随机游动标度极限构造的粗糙超布朗运动的紧支撑性质，通过内部估计方法克服了奇异 SPDE 无法使用经典 PDE 论证的困难，证明了该性质依然成立。
In this work we study fractal properties of rough differential equations driven by a fractional Brownian motions with Hurst parameter $H>\frac{1}{4}$. In particular, we show that the Hausdorff dimension of the sample paths of the solution is $\min\{d,\frac{1}{H}\}$ and that the Hausdorff dimension of the level set $L_x=\{ t\in[ε,1]: X_t=x\}$ is $1-dH$ with positive probability when $d<\frac{1}{H}$
We construct a $K$-rough path above either a space-time or a spatial fractional Brownian motion, in any space dimension $d$. This allows us to provide an interpretation and a unique solution for the corresponding parabolic Anderson model, understood in the renormalized sense. We also consider the case of a spatial fractional noise.
We consider two related linear PDE's perturbed by a fractional Brownian motion. We allow the drift to be discontinuous, in which case the corresponding deterministic equation is ill-posed. However, the noise will be shown to have a regularizing effect on the equations in the sense that we can prove existence of solutions for almost all paths of the fractional Brownian motion.
We give meaning to differential equations with a rough path term and a Brownian noise term as driving signals. Such differential equations as well as the question of regularity of the solution map arise naturally and we discuss two applications: one revisits Clark's robustness problem in nonlinear filtering, the other is a Feynman--Kac type representation of linear RPDEs. En passant, we give a short and direct argument that implies integrability estimates for rough differential equations with Gaussian driving signals which is of independent interest.
We study a stochastic differential equation in the sense of rough path theory driven by fractional Brownian rough path with Hurst parameter H (1/3 < H <= 1/2) under the ellipticity assumption at the starting point. In such a case, the law of the solution at a fixed time has a kernel, i.e., a density function with respect to Lebesgue measure. In this paper we prove a short time off-diagonal asymptotic expansion of the kernel under mild additional assumptions. Our main tool is Watanabe's distributional Malliavin calculus.
This note is devoted to construct a rough path above a multidimensional fractional Brownian motion $B$ with any Hurst parameter $H\in(0,1)$, by means of its representation as a Volterra Gaussian process. This approach yields some algebraic and computational simplifications with respect to [Stochastic Process. Appl. 120 (2010) 1444--1472], where the construction of a rough path over $B$ was first introduced.
We study the convergence in rough path topology of a certain class of discrete processes, the hidden Markov walks, to a Brownian motion with an area anomaly. This area anomaly, which is a new object, keeps track of the time-correlation of the discrete models and brings into light the question of embeddings of discrete processes into continuous time. We also identify an underlying combinatorial structure in the hidden Markov walks, which turns out to be a generalization of the occupation time from the classical ergodic theorem in the spirit of rough paths.
We study the scaling limit of a branching random walk in static random environment in dimension $d=1,2$ and show that it is given by a super-Brownian motion in a white noise potential. In dimension $1$ we characterize the limit as the unique weak solution to the stochastic PDE: \[\partial_t μ= (Δ{+} ξ) μ{+} \sqrt{2νμ} \tildeξ\] for independent space white noise $ξ$ and space-time white noise $\tildeξ$. In dimension $2$ the study requires paracontrolled theory and the limit process is described via a martingale problem. In both dimensions we prove persistence of this rough version of the super-…
A Generalized Rough Super Brownian Motion
本文构造了随机环境中后代分布具有无限方差的分支随机游走的标度极限，证明其拉普拉斯泛函由非线性PAM给出，并建立了该超过程的鞅刻画、紧支撑性质与指数持续性。
The indefinite integral of the homogenized Ornstein-Uhlenbeck process is a well-known model for physical Brownian motion, modelling the behaviour of an object subject to random impulses [L. S. Ornstein, G. E. Uhlenbeck: On the theory of Brownian Motion. In: Physical Review. 36, 1930, 823-841]. One can scale these models by changing the mass of the particle and in the small mass limit one has almost sure uniform convergence in distribution to the standard idealized model of mathematical Brownian motion. This provides one well known way of realising the Wiener process. However, this result is le…
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…
Regularisation by Gaussian rough path lifts of fractional Brownian motions
本文证明了由分数布朗运动高斯粗糙路径提升驱动的分布漂移粗糙微分方程的随机强适定性，在 Hurst 参数 $H\in(1/3,1/2)$ 下给出了 Catellier-Gubinelli 结果的乘法类比。
Fourier normal ordering \cite{Unt09bis} is a new algorithm to construct explicit rough paths over arbitrary Hölder-continuous multidimensional paths. We apply in this article the Fourier normal ordering ordering algorithm to the construction of an explicit rough path over multi-dimensional fractional Brownian motion $B$ with arbitrary Hurst index $α$ (in particular, for $α\le 1/4$, which was till now an open problem) by regularizing the iterated integrals of the analytic approximation of $B$ defined in \cite{Unt08}. The regularization procedure is applied to 'Fourier normal ordered' iterated i…
In this paper we derive a Bismut-Elworthy-Li type formula with respect to strong solutions to singular stochastic differential equations (SDE's) with additive noise given by a multi-dimensional fractional Brownian motion with Hurst parameter $H<1/2$. "Singular" here means that the drift vector field of such equations is allowed to be merely bounded and integrable. As an application we use this representation formula for the study of price sensitivities of financial claims based on a stock price model with stochastic volatility, whose dynamics is described by means of fractional Brownian motion…
In this article, we consider the so-called modified Euler scheme for stochastic differential equations (SDEs) driven by fractional Brownian motions (fBm) with Hurst parameter $\frac13<H<\frac12$. This is a first-order time-discrete numerical approximation scheme, and has been recently introduced by Hu, Liu and Nualart in order to generalize the classical Euler scheme for Itô SDEs to the case $H>\frac12$. The current contribution generalizes the modified Euler scheme to the rough case $\frac13<H<\frac12$. Namely, we show a convergence rate of order $n^{\frac12-2H}$ for the scheme, and we argue …
This paper is devoted to studying the averaging principle for fast-slow system of rough differential equations driven by mixed fractional Brownian rough path. The fast component is driven by Brownian motion, while the slow component is driven by fractional Brownian motion with Hurst index $H ~(1/3 < H\leq 1/2)$. Combining the fractional calculus approach to rough path theory and Khasminskii's classical time discretization method, we prove that the slow component strongly converges to the solution of the corresponding averaged equation in the $L^1$-sense. The averaging principle for a fast-slow…
Rough volatility models are becoming increasingly popular in quantitative finance. In this framework, one considers that the behavior of the log-volatility process of a financial asset is close to that of a fractional Brownian motion with Hurst parameter around 0.1. Motivated by this, we wish to define a natural and relevant limit for the fractional Brownian motion when $H$ goes to zero. We show that once properly normalized, the fractional Brownian motion converges to a Gaussian random distribution which is very close to a log-correlated random field.
The original Donsker theorem says that a standard random walk converges in distribution to a Brownian motion in the space of continuous functions. It has recently been extended to enriched random walks and enriched Brownian motion. We use the Stein-Dirichlet method to precise the rate of this convergence in the topology of fractional Sobolev spaces.
Stochastic Calculus for Rough Fractional Brownian Motion via Operator Factorization
本文提出了一种基于算子分解的粗糙分数布朗运动随机微积分框架，通过高斯Volterra过程的规范分解，建立了显式导数公式和条件期望展开，并将Gubinelli导数识别为可预测成分，为混合半鞅-粗糙过程提供统一的微积分方法。
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。
This article is concerned with stochastic differential equations driven by a $d$ dimensional fractional Brownian motion with Hurst parameter $H>1/4$, understood in the rough paths sense. Whenever the coefficients of the equation satisfy a uniform hypoellipticity condition, we establish a sharp local estimate on the associated control distance function and a sharp local lower estimate on the density of the solution. Our methodology relies heavily on the rough paths structure of the equation.
Super and Weak Poincaré Inequalities for Sticky-Reflected Diffusion Processes
本文为粘性反射扩散过程建立超和弱Poincaré不等式，并刻画相关扩散半群的收敛速率和一致可积性。
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…
Pathwise Uniqueness for Multiplicative Young and Rough Differential Equations Driven by Fractional Brownian Motion
该研究证明了任意维度下由赫斯特参数$H\in(1/3,1)$的分数布朗运动驱动的乘性随机微分方程的轨道唯一性，通过结合随机平均估计与改进版随机缝合引理，放宽了波动率系数的赫尔德连续阶数要求，改进了已有经典结果。
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 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 consider a complex-valued d-dimensional fractional Brownian motion defined on the closure of the complex upper half-plane, called analytic fractional Brownian motion. This process has been introduced by the second author of the article, and both its real and imaginary parts, restricted on the real axis, are usual fractional Brownian motions. The current note is devoted to prove that a rough path based on the analytic fBm can be constructed for any value of the Hurst parameter in (0,1/2). This allows in particular to solve differential equations driven by this process in a nei…
Fractional Brownian motion with the Hurst parameter $H<\frac{1}{2}$ is used widely, for instance, to describe a 'rough' stochastic volatility process in finance. In this paper, we examine an Ait-Sahalia-type interest rate model driven by a fractional Brownian motion with $H<\frac{1}{2}$ and establish theoretical properties such as an existence-and-uniqueness theorem, regularity in the sense of Malliavin differentiability and higher moments of the strong solutions.
We investigate the optimal Hölder continuity and hitting probabilities for systems of stochastic heat equations and stochastic wave equations driven by an additive fractional Brownian sheet with temporal index $1/2$ and spatial index $H\le1/2$. Using stochastic calculus for fractional Brownian motion, we prove that these systems are well-posed and the solutions are Hölder continuous. Furthermore, the optimal Hölder exponents are obtained, which is the first result, as far as we knew, on the optimal Hölder continuity of SHEs and SWEs driven by fractional Brownian sheet that is rough in space. B…
In 1990, in Itô's stochastic calculus framework, Aubin and Da Prato established a necessary and sufficient condition of invariance of a nonempty compact or convex subset $C$ of $\mathbb R^d$ ($d\in\mathbb N^*$) for stochastic differential equations (SDE) driven by a Brownian motion. In Lyons rough paths framework, this paper deals with an extension of Aubin and Da Prato's results to rough differential equations. A comparison theorem is provided, and the special case of differential equations driven by a fractional Brownian motion is detailed.
We consider stochastic differential equations dY=V(Y)dX driven by a multidimensional Gaussian process X in the rough path sense. Using Malliavin Calculus we show that Y(t) admits a density for t in (0,T] provided (i) the vector fields V=(V_1,...,V_d) satisfy Hoermander's condition and (ii) the Gaussian driving signal X satisfies certain conditions. Examples of driving signals include fractional Brownian motion with Hurst parameter H>1/4, the Brownian Bridge returning to zero after time T and the Ornstein-Uhlenbeck process.
Robustness of Optimal Controlled Diffusions with Near-Brownian Noise via Rough Paths Theory
本文利用粗糙路径理论证明受控随机微分方程对近 Brown 噪声的鲁棒性，表明 Brown 模型下的最优 Lipschitz 控制策略在非 Brown 近似噪声系统中仍近似最优。
The maximum likelihood type estimator of SDEs with fractional Brownian motion under small noise asymptotics in the rough case
针对分数布朗运动驱动的随机微分方程，在漂移和扩散系数满足一定假设时，构造最大似然估计量，证明小噪声下漂移参数的渐近正态性与矩收敛。
We consider a differential equation driven by a Brownian motion as well as a rough path. We prove a Girsanov-type result for this equation to construct a weak solution in the probabilistic sense.
We show that the tail probability of the rough line integral $\int_{0}^{1}ϕ(X_{t})dY_{t}$, where $(X,Y)$ is a 2D fractional Brownian motion with Hurst parameter $H\in(1/4,1/2)$ and $ϕ$ is a $C_{b}^{\infty}$-function satisfying a mild non-degeneracy condition on its derivative, cannot decay faster than a $γ$-Weibull tail with any exponent $γ>2H+1$. In particular, this produces a simple class of examples of differential equations driven by fBM, whose solutions fail to have Gaussian tail even though the underlying vector fields are assumed to be of class $C_{b}^{\infty}$. This also demonstrates t…
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。
As a general rule, differential equations driven by a multi-dimensional irregular path $Γ$ are solved by constructing a rough path over $Γ$. The domain of definition ? and also estimates ? of the solutions depend on upper bounds for the rough path; these general, deterministic estimates are too crude to apply e.g. to the solutions of stochastic differential equations with linear coefficients driven by a Gaussian process with Hölder regularity $α< 1/2$. We prove here (by showing convergence of Chen's series) that linear stochastic differential equations driven by analytic fractional Brownian mo…
This article is concerned with stochastic differential equations driven by a $d$ dimensional fractional Brownian motion with Hurst parameter $H>1/4$, understood in the rough paths sense. Whenever the coefficients of the equation satisfy a uniform hypoellipticity condition, we establish a sharp local estimate on the associated control distance function and a sharp local lower estimate on the density of the solution. Our methodology relies heavily on the rough paths structure of the equation.
The main tool for stochastic calculus with respect to a multidimensional process $B$ with small Hölder regularity index is rough path theory. Once $B$ has been lifted to a rough path, a stochastic calculus -- as well as solutions to stochastic differential equations driven by $B$ -- follow by standard arguments. Although such a lift has been proved to exist by abstract arguments \cite{LyoVic07}, a first general, explicit construction has been proposed in \cite{Unt09,Unt09bis} under the name of Fourier normal ordering. The purpose of this short note is to convey the main ideas of the Fourier no…
Geometric Rough Paths above Mixed Fractional Brownian Motion
本文研究了混合分数布朗运动的几何粗糙路径理论，建立了其与马尔可夫算子框架的联系，并拓展了经典结果至混合情况。
The $d$-dimensional fractional Brownian motion (FBM for short) $B_t=((B_t^{(1)},...,B_t^{(d)}),t\in\mathbb{R})$ with Hurst exponent $α$, $α\in(0,1)$, is a $d$-dimensional centered, self-similar Gaussian process with covariance ${\mathbb{E}}[B_s^{(i)}B _t^{(j)}]={1/2}δ_{i,j}(|s|^{2α}+|t|^{2α}-|t-s|^{2 α}).$ The long-standing problem of defining a stochastic integration with respect to FBM (and the related problem of solving stochastic differential equations driven by FBM) has been addressed successfully by several different methods, although in each case with a restriction on the range of eithe…
Rough backward stochastic differential equations
本文提出一种直接基于粗糙半鞅的粗糙倒向随机微分方程理论，在正则性条件下建立适定性，并建立与倒向双随机微分方程的对应关系。
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.
We prove a representation for the support of McKean Vlasov Equations. To do so, we construct functional quantizations for the law of Brownian motion as a measure over the (non-reflexive) Banach space of Hölder continuous paths. By solving optimal Karhunen Loève expansions and exploiting the compact embedding of Gaussian measures, we obtain a sequence of deterministic finite supported measures that converge to the law of a Brownian motion with explicit rate. We show the approximation sequence is near optimal with very favourable integrability properties and prove these approximations remain tru…
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