Marco Romito
We extend the validity of a simple method for the existence of a density for stochastic differential equations, first introduced in [DebRom2014], by proving local estimate for the density, existence for the density with summable drift, and by improving the regularity of the density.
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Rough Stochastic Analysis with Jumps
本文提出新的粗糙随机缝合引理，用于处理多重不连续控制函数，进而发展了在 càdlàg 设定下的粗糙随机分析理论，定义了粗糙随机积分并建立了带有跳跃不连续噪声的随机微分方程解及带跳跃的随机控制路径的伊藤公式。
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.
In this note we introduce a new approach to rough and stochastic partial differential equations (RPDEs and SPDEs): we consider general Banach spaces as state spaces and -- for the sake of simiplicity -- finite dimensional sources of noise, either rough or stochastic. By means of a time-dependent transformation of state space and rough path theory we are able to construct unique solutions of the respective R- and SPDEs. As a consequence of our construction we can apply the pool of results of rough path theory, in particular we obtain strong and weak numerical schemes of high order converging to…
We investigate mild solutions for stochastic evolution equations driven by a fractional Brownian motion (fBm) with Hurst parameter H in (1/3, 1/2] in infinite-dimensional Banach spaces. Using elements from rough paths theory we introduce an appropriate integral with respect to the fBm. This allows us to solve pathwise our stochastic evolution equation in a suitable function space.
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。
In this paper we establish a Taylor-like expansion in the context of the rough path theory for a family of It ^{o} maps indexed by a small parameter. We treat not only the case that the roughness $p$ satisfies $[p]=2$, but also the case that $[p] ge 3$. As an application, we discuss the Laplace asymptotics for It^{o} functionals of Brownian rough paths.
Controlled fields, rough stochastic calculus, and Itô-Wentzell-Alekseev-Gröbner identities
本文提出了一种用于粗糙随机系统的时空受控场微积分，推导了粗糙随机伊藤-文兹尔公式，并基于前人工作建立了相关恒等式。
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。
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.
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。
An integrable bound for semilinear rough partial differential equations with unbounded diffusion coefficients
本文研究了半线性粗糙偏微分方程解的受控粗糙路径范数的矩界，创新性地处理了低时间正则性粗糙路径和无界扩散系数，并基于Banach空间的单调尺度与控制函数引入了新的受控粗糙路径概念。
Multidimensional Backward Stochastic Differential Equations with Rough Drifts
本文研究具有粗糙漂移的多维倒向随机微分方程，通过引入 $p$-粗糙随机积分并利用不动点论证，证明了在特定条件下适应解的存在唯一性，并建立了其与粗糙偏微分方程组的联系。
Existence and uniqueness results for strongly degenerate McKean-Vlasov equations with rough coefficients
本文针对具粗糙系数的强退化McKean-Vlasov方程，扩展并改进了已有技术以证明弱解的存在性，并在特定结构条件下建立了弱与强适定性结果。
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 […
This paper studies the stochastic heat equation driven by time fractional Gaussian noise with Hurst parameter $H\in(0,1/2)$. We establish the Feynman-Kac representation of the solution and use this representation to obtain matching lower and upper bounds for the $L^p(Ω)$ moments of the solution.
In this paper, we obtain the existence of random attractors for a class of evolution equations driven by a geometric fractional Brownian rough path with Hurst index $H\in(\frac{1}{3},\frac{1}{2}]$ and establish the upper semi-continuity of random attractors $\mathcal{A}_η$ for the approximated systems of the evolution equations.
We 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…
Non-explosion principles for branched rough differential equations with unbounded coefficients
针对带无界漂移和粗糙系数的分支粗糙微分方程，提出非爆炸判据，通过刻画零路径上的纯面积分支粗糙路径构造爆炸反例证明其尖锐性，并利用系数增长与高阶导数衰减的权衡给出更优的非爆炸原理。
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.
We highlight a fundamental ill-posedness issue for nonlinear stochastic wave equations driven by a fractional noise. Namely, if the noise becomes too rough (i.e., the sum of its Hurst indexes becomes too small), then there is essentially no hope to provide an interpretation of the model, whether directly or through a Wick-type renormalization procedure. This phenomenon can be compared with the situation of a general SDE driven by a two-dimensional fractional noise of index $H\leq \frac{1}{4}$. Our results clarify and extend previous similar properties exhibited in [3] or in [14].
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--…
Analytically weak solutions to stochastic heat equations with spatially rough noise
本文提出一种利用变分方法或Krylov的$L^p$-理论在解析弱意义下求解空间粗糙噪声驱动的随机热方程的简单替代方法，并得出多项改进推论。
Bi-parameter local linearization for the stochastic wave equation with rough noise
本文研究由时间白噪声和空间粗糙噪声驱动的一维非线性随机波动方程，证明双参数局部线性化，并据此建立二次变差极限和扩散参数的一致估计量，数值实验验证了其性能。
In this paper we review and improve pathwise uniqueness results for some types of one-dimensional stochastic differential equations (SDE) involving the local time of the unknown process. The diffusion coefficient of the SDEs we consider is allowed to vanish on a set of positive measure and is not assumed to be smooth. As opposed to various existing results, our arguments are mainly based on the comparison theorem for local time and the occupation time formula. We apply our pathwise uniqueness results to derive strong existence and other properties of solutions for SDEs with rough coefficients.
In this paper, we consider a class of slow-fast systems of stochastic partial differential equations where the nonlinearity in the slow equation is not continuous and unbounded. We first provide conditions that ensure the existence of a martingale solution. Then we prove that the laws of the slow motions are tight, and any of their limiting points is a martingale solution for a suitable averaged equation. Our results apply to systems of stochastic reaction-diffusion equations where the reaction term in the slow equation is only continuous and has polynomial growth.
We study strong existence and pathwise uniqueness for stochastic differential equations in $\RR^d$ with rough coefficients, and without assuming uniform ellipticity for the diffusion matrix. Our approach relies on direct quantitative estimates on solutions to the SDE, assuming Sobolev bounds on the drift and diffusion coefficients, and $L^p$ bounds for the solution of the corresponding Fokker-Planck PDE, which can be proved separately. This allows a great flexibility regarding the method employed to obtain these last bounds. Hence we are able to obtain general criteria in various cases, includ…
Mild Solution of Semilinear Rough Stochastic Evolution Equations
针对带线性粗糙项的半线性随机抛物方程，采用随机受控粗糙路径方法与不动点论证证明温和解的存在唯一性，并提出适用于单调族过程的温和随机缝补引理作为定义粗糙随机卷积的技术工具。
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 article, we consider fractional stochastic wave equations on $\mathbb R$ driven by a multiplicative Gaussian noise which is white/colored in time and has the covariance of a fractional Brownian motion with Hurst parameter $H\in(\frac14, \frac12)$ in space. We prove the existence and uniqueness of the mild Skorohod solution, establish lower and upper bounds for the $p$-th moment of the solution for all $p\ge2$, and obtain the Hölder continuity in time and space variables for the solution.
Controlled rough SDEs, pathwise stochastic control and dynamic programming principles
本文研究粗糙随机微分方程的最优控制，建立路径动态规划原理与HJB稳定性，无需布朗统计，统一不同控制类别。
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。
Rough stochastic filtering
本文提出一种新的混合粗糙随机方法，解决非线性随机过滤方程的适定性问题，并推导出粗糙过滤方程，提供了一种稳健且统一的解决方案。
In this article, we consider the following stochastic fractional diffusion equation \begin{equation*} \left(\partial^β+\dfracν{2}\left(-Δ\right)^{α/ 2}\right) u(t, x)= λ\: I_{0_+}^γ\left[u(t, x) \dot{W}(t, x)\right] ,\quad t>0,\: x \in \mathbb{R}, \end{equation*} where $α>0$, $β\in(0,2]$, $γ\ge 0$, $λ\neq0$, $ν>0$, and $\dot{W}$ is a Gaussian noise which is white or fractional in time and rough in space. We prove the existence and uniqueness of the solution in the Itô-Skorohod sense and obtain the lower and upper bounds for the $p$-th moment. The Hölder regularity of the solution is also studi…
Pontryagin Maximum Principle for rough stochastic systems and pathwise stochastic control
本文提出了一种用于粗糙随机系统和路径随机控制问题的最大原理，证明了该问题与路径随机控制问题具有相同的价值函数，并展示了线性二次问题的最优控制结果。
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.
In this note we consider the parabolic Anderson model in one dimension with time-independent fractional noise $\dot{W}$ in space. We consider the case $H<\frac{1}{2}$ and get existence and uniqueness of solution. In order to find the quenched asymptotics for the solution we consider its Feynman-Kac representation and explore the asymptotics of the principal eigenvalue for a random operator of the form $\frac{1}{2} Δ+ \dot{W}$.
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…
Malliavin Calculus for rough stochastic differential equations
本文证明粗糙随机微分方程是 Malliavin 可微的，并在椭圆性或 Hörmander 条件下证明解密度的存在性与光滑性，关键发展了线性 RSDE 理论。
We consider long-time behavior of dynamical systems perturbed by a small noise. Under certain conditions, a slow component of such a motion, which is most important for long- time evolution, can be described as a motion on the cone of invariant measures of the non- perturbed system. The case of a finite number of extreme points of the cone is considered in this paper. As is known, in the generic case, the long-time evolution can be described by a hierarchy of cycles defined by the action functional for corresponding stochastic processes. This, in particular, allows to study metastable distribu…
Rough SDEs and Robust Filtering for Jump-Diffusions
研究多维相关跳跃扩散模型中条件分布的鲁棒（连续）表示，通过分析带跳跃的粗糙随机微分方程，建立了随机滤波的若干新鲁棒性结果。
Stochastic nonlinear wave equation with rougher than white noise
本文研究了 $\mathbb T^2$ 上由高斯粗糙 Matérn 噪声驱动的三次奇异随机波动方程，通过在 Bourgain 空间中发展新的三线性估计和三次计数估计，证明了 $α< \tfrac{3}{8}$ 时的局部适定性，推广并加强了已有结果。
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.
We study the Taylor expansion for the solutions of differential equations driven by $p$-rough paths with $p>2$. We prove a general theorem concerning the convergence of the Taylor expansion on a nonempty interval provided that the vector fields are analytic on a ball centered at the initial point. We also derive criteria that enable us to study the rate of convergence of the Taylor expansion. Finally and this is also the main and the most original part of this paper, we prove Castell expansions and tail estimates with exponential decays for the remainder terms of the solutions of the stochasti…
This paper is devoted to investigating Freidlin-Wentzell's large deviation principle for one (spatial) dimensional nonlinear stochastic wave equation $\frac{\partial^2 u^{\e}(t,x)}{\partial t^2}=\frac{\partial^2 u^{\e}(t,x)}{\partial x^2}+\sqrt{\e}σ(t, x, u^{\e}(t,x))\dot{W}(t,x)$, where $\dot{W}$ is white in time and fractional in space with Hurst parameter $H\in(\frac 14,\frac 12)$. The variational framework and the modified weak convergence criterion proposed by Matoussi et al. \cite{MSZ} are adopted here.
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。
In this paper, we are concerned with convergence rate of Euler-Maruyama scheme for stochastic differential equations with rough coefficients. The key contributions lie in (i), by means of regularity of non-degenerate Kolmogrov equation, we investigate convergence rate of Euler-Maruyama scheme for a class of stochastic differential equations, which allow the drifts to be Dini-continuous and unbounded; (ii) by the aid of regularization properties of degenerate Kolmogrov equation, we discuss convergence rate of Euler-Maruyama scheme for a range of degenerate stochastic differential equations, whe…
We study space-time regularity of the solution of the nonlinear stochastic heat equation in one spatial dimension driven by space-time white noise, with a rough initial condition. This initial condition is a locally finite measure $μ$ with, possibly, exponentially growing tails. We show how this regularity depends, in a neighborhood of $t=0$, on the regularity of the initial condition. On compact sets in which $t>0$, the classical Hölder-continuity exponents $\frac{1}{4}-$ in time and $\frac{1}{2}-$ in space remain valid. However, on compact sets that include $t=0$, the Hölder continuity of th…
We define and solve Volterra equations driven by an irregular signal, by means of a variant of the rough path theory allowing to handle generalized integrals weighted by an exponential coefficient. The results are applied to the fractional Brownian motion with Hurst coefficient greater than 1/3
We study fully nonlinear second-order (forward) stochastic partial differential equations (SPDEs). They can also be viewed as forward path-dependent PDEs (PPDEs) and will be treated as rough PDEs (RPDEs) under a unified framework. We develop first a local theory of classical solutions and define then viscosity solutions through smooth test functions. Our notion of viscosity solutions is equivalent to the alternative one using semi-jets. Next, we prove basic properties such as consistency, stability, and a partial comparison principle in the general setting. When the diffusion coefficient is se…
Parameter dependent rough SDEs with applications to rough PDEs
本文研究参数依赖的粗糙 SDE 的 \mathscr{L}-可微性，用以建立粗糙 Kolmogorov 倒向方程正则解的适定性，并为去除相关随机滤波方程的维度依赖正则性假设奠定基础。
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.
In rough stochastic PDE theory of Hairer type, rough path lifts with respect to the space variable of two-parameter continuous Gaussian processes play a main role. A prominent example of such processes is the solution of the stochastic heat equation under the periodic condition. The main objective of this paper is to show that the law of the spatial lift of this process satisfies a Schilder type large deviation principle on the continuous path space over a geometric rough path space.
We prove the existence and uniqueness of solutions to a class of stochastic scalar conservation laws with joint space-time transport noise and affine-linear noise driven by a geometric p-rough path. In particular, stability of the solutions with respect to the driving rough path is obtained, leading to a robust approach to stochastic scalar conservation laws. As immediate corollaries we obtain support theorems, large deviation results and the generation of a random dynamical system.
This paper studies the nonlinear one-dimensional stochastic heat equation driven by a Gaussian noise which is white in time and which has the covariance of a fractional Brownian motion with Hurst parameter 1/4\textless{}H\textless{}1/2 in the space variable. The existence and uniqueness of the solution u are proved assuming the nonlinear coefficient is differentiable with a Lipschitz derivative and vanishes at 0. In the case of a multiplicative noise, that is the linear equation, we derive the Wiener chaos expansion of the solution and a Feynman-Kac formula for the moments of the solution. The…
McKean-Vlasov equations with rough common noise
本文在作者近期提出的粗糙随机微分方程框架下，证明了带粗糙公共噪声和循序可测系数的McKean-Vlasov方程的适定性，其正则性假设与Itô及粗糙路径理论要求一致。
An integrable bound for rough stochastic partial differential equations with applications to invariant manifolds and stability
本文针对由 Gaussian 过程驱动的半线性粗糙随机偏微分方程，建立了 $\mathcal{L}^p(Ω)$-可积先验界，并借助 Banach 空间乘法遍历定理得到平稳点附近的 Lyapunov 谱、局部不变流形及指数稳定性结果。
Motivated by the recent advances in the theory of stochastic partial differential equations involving nonlinear functions of distributions, like the Kardar-Parisi-Zhang (KPZ) equation, we reconsider the unique solvability of one-dimensional stochastic differential equations, the drift of which is a distribution, by means of rough paths theory. Existence and uniqueness are established in the weak sense when the drift reads as the derivative of a H{ö}lder continuous function. Regularity of the drift part is investigated carefully and a related stochastic calculus is also proposed, which makes th…
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…
We investigate the problem of the rate of convergence to equilibrium for ergodic stochastic differential equations driven by fractional Brownian motion with Hurst parameter $H\in (1/3,1)$ and multiplicative noise component $σ$. When $σ$ is constant and for every $H\in (0,1)$, it was proved in [19] that, under some mean-reverting assumptions, such a process converges to its equilibrium at a rate of order $t^{-α}$ where $α\in (0,1)$ (depending on $H$). In [11], this result has been extended to the multiplicative case when $H\textgreater{}1/2$. In this paper, we obtain these types of results in t…
Stochastic differential equations (SDEs) on compact foliated spaces were introduced a few years ago. As a corollary, a leafwise Brownian motion on a compact foliated space was obtained as a solution to an SDE. In this paper we construct stochastic flows associated with the SDEs by using rough path theory, which is something like a "deterministic version" of Itô's SDE theory.
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