Yuzuru Inahama
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.
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The $L^q$-bounds for Derivatives of Unitary Developments of Random Continuous Geometric Rough Paths
本文推导随机连续几何粗糙路径酉发展导数的显式公式并建立$L^q$界，定量研究特征函数解析性，证明在温和条件下签名分布由期望签名决定，部分回答了Lyons和Ni的开放问题。
Functional quantization of rough volatility and applications to volatility derivatives
本研究提出粗糙波动率的乘积泛函量化方法，其量化器可离线计算，在粗糙波动率数值工具领域具备竞争力，且在粗糙Bergomi模型的VIX期货定价上与现有基准方法表现相当。
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。
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。
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。
We introduce a differential structure for the space of weakly geometric p rough paths over a Banach space V for 2<p<3. We begin by considering a certain natural family of smooth rough paths and differentiating in the truncated tensor series. The resulting object has a clear interpretation, even for non-smooth rough paths, which we take to be an element of the tangent space. We can associate it uniquely to an equivalence class of curves, with equivalence defined by our differential structure. Thus, for a functional on rough path space, we can define the derivative in a tangent direction analogo…
Rough paths and symmetric-Stratonovich integrals driven by singular covariance gaussian processes
本文研究了随机粗糙路径积分与对称-Stratonovich积分的关系，在Malliavin calculus温和条件下证明了两者等价，并得到了一阶Stratonovich方案的几乎必然收敛速率。
According to DiPerna-Lions theory, velocity fields with weak derivatives in $L^p$ spaces possess weakly regular flows. When a velocity field is perturbed by a white noise, the corresponding (stochastic) flow is far more regular in spatial variables; a $d$-dimensional diffusion with a drift in $L^{r,q}$ space ($r$ for the spatial variable and $q$ for the temporal variable) possesses weak derivatives with stretched exponential bounds, provided that $r/d+2/q<1$. As an application we show that a Hamiltonian system that is perturbed by a white noise produces a symplectic flow provided that the corr…
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 derive an invariance principle for the lift to the rough path topology of stochastic processes with delayed regenerative increments under an optimal moment condition. An interesting feature of the result is the emergence of area anomaly, a correction term in the second level of the limiting rough path which is identified as the average stochastic area on a regeneration interval. A few applications include random walks in random environment and additive functionals of recurrent Markov chains. The result is formulated in the p-variation settings, where a rough Donsker Theorem is available und…
Moments in Rough Bergomi and Boundary Attainment in Rough Heston
本文证明了粗糙Bergomi价格过程在负相关下整个次临界范围内的有限正矩，并证明了粗糙Heston方差过程在任意正时间以正概率到达零边界。
We calculate joint moments of the characteristic polynomial of a random unitary matrix from the circular unitary ensemble and its derivative in the case that the power in the moments is an odd positive integer. The calculations are carried out for finite matrix size and in the limit as the size of the matrices goes to infinity. The latter asymptotic calculation allows us to prove a long-standing conjecture from random matrix theory.
We study linear rough partial differential equations in the setting of [Friz and Hairer, Springer, 2014, Chapter 12]. More precisely, we consider a linear parabolic partial differential equation driven by a deterministic rough path $\mathbf{W}$ of Hölder regularity $α$ with $1/3 < α \le 1/2$. Based on a stochastic representation of the solution of the rough partial differential equation, we propose a regression Monte Carlo algorithm for spatio-temporal approximation of the solution. We provide a full convergence analysis of the proposed approximation method which essentially relies on the new …
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 define a characteristic function for probability measures on the signatures of geometric rough paths. We determine sufficient conditions under which a random variable is uniquely determined by its expected signature, thus partially solving the analogue of the moment problem. We furthermore study analyticity properties of the characteristic function and prove a method of moments for weak convergence of random variables. We apply our results to signature arising from Lévy, Gaussian and Markovian rough paths.
In this paper, we consider the KPZ equation driven by space-time white noise replaced with its fractional derivatives of order $γ>0$ in spatial variable. A well-posedness theory for the KPZ equation is established by Hairer [3] as an application of the theory of regularity structures. Our aim is to see to what extent his theory works if noises become rougher. We can expect that his theory works if and only if $γ<1/2$. However, we show that the renormalization like "$(\partial_x h)^2-\infty$" is well-posed only if $γ<1/4$.
An example driven introduction to probabilistic rough paths
本文介绍了用于求解粗糙平均场方程、以Lions树为索引集的新型正则性结构，该结构源自含Lions导数的泰勒展开，是对已有思想的全面推广，可助力研究集体相互作用对个体动力学的影响。
Spatial differentiability of solutions of stochastic differential equations (SDEs) is a classical question in stochastic analysis. The case of coefficients with globally Lipschitz continuous derivatives is well understood in the literature. Counterexamples with smooth and bounded coefficients demonstrate that the non-globally Lipschitz case is more subtle. In this article we establish conditions, including a suitable local monotonicity property, which provide existence of continuously differentiable solutions of SDEs, moment estimates and strong local Hölder regularity.
In this paper, we study the parabolic Anderson model of Skorohod type driven by a fractional Gaussian noise in time with Hurst parameter $H \in (0, 1/2)$. By using the Feynman-Kac representation for the $L^p(Ω)$ moments of the solution, we find the upper and lower bounds for the moments.
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…
$\mathfrak{X}$PDE for $\mathfrak{X} \in \{\mathrm{BS},\mathrm{FBS}, \mathrm{P}\}$: a rough volatility context
本文在粗糙波动率背景下精确建立了路径依赖偏微分方程与倒向随机偏微分方程之间的联系，并将 Bayer、Qiu 和 Yao 引入的随机场识别为价值函数的逐路径导数。
A solution of the heat equation with a distribution-valued potential is constructed by regularization. When the potential is the generalized derivative of a Hölder continuous function, regularity of the resulting solution is in line with the standard parabolic theory.
Systems of rough stochastic differential equations I: weak existence and Yamada--Watanabe
本文证明粗糙随机微分方程系统在无界系数下的弱存在性，并建立 Yamada--Watanabe 定理：弱存在性与路径唯一性蕴含强解及联合分布唯一性。
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 present a number of related comparison results, which allow to compare moment explosion times, moment generating functions and critical moments between rough and non-rough Heston models of stochastic volatility. All results are based on a comparison principle for certain non-linear Volterra integral equations. Our upper bound for the moment explosion time is different from the bound introduced by Gerhold, Gerstenecker and Pinter (2018) and tighter for typical parameter values. The results can be directly transferred to a comparison principle for the asymptotic slope of implied volatility be…
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…
The purpose of this paper is to establish a Feynman-Kac formula for the moments of the iterated Malliavin derivatives of the solution to the parabolic Anderson model in terms of pinned Brownian motions. As an application, we obtain estimates for the moments of the iterated derivatives of the solution.
Annealed functional CLT in the rough path topology is proved for the standard class of ballistic random walks in random environment. Moreover, the `area anomaly', i.e. a deterministic linear correction for the second level iterated integral of the rescaled path, is identified in terms of a stochastic area on a regeneration interval. The main theorem is formulated in more general settings, namely for any discrete process with uniformly bounded increments which admits a regeneration structure where the regeneration times have finite moments. Here the largest finite moment translates into the deg…
We study the nonlinear stochastic heat equation in the spatial domain $\mathbb {R}$, driven by space-time white noise. A central special case is the parabolic Anderson model. The initial condition is taken to be a measure on $\mathbb {R}$, such as the Dirac delta function, but this measure may also have noncompact support and even be nontempered (e.g., with exponentially growing tails). Existence and uniqueness of a random field solution is proved without appealing to Gronwall's lemma, by keeping tight control over moments in the Picard iteration scheme. Upper bounds on all $p$th moments $(p\g…
We study the number of local maxima with given radial derivative of spherical mixed $p$-spin models and prove that the second moment matches the square of the first moment on exponential scale for arbitrary mixtures and any radial derivative. This is surprising, since for the number of local maxima with given radial derivative and given energy the corresponding result is only true for specific mixtures [Sub17; BSZ20]. We use standard Kac-Rice computations to derive formulas for the first and second moment at exponential scale, and then find a remarkable analytic argument that shows that the se…
Parabolic stochastic PDEs on bounded domains with rough initial conditions: moment and correlation bounds
本文研究了有界Lipschiz区域上由Gaussian噪声驱动的非线性抛物型随机偏微分方程，在测度值初始数据下建立了解的存在性、唯一性、矩界与两点相关函数的显式上下界，并证明了解在特定噪声水平下的完全间歇性。
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…
This paper studies the one-dimensional parabolic Anderson model driven by a Gaussian noise which is white in time and has the covariance of a fractional Brownian motion with Hurst parameter $H \in (\frac{1}{4}, \frac{1}{2})$ in the space variable. We derive the Wiener chaos expansion of the solution and a Feynman-Kac formula for the moments of the solution. These results allow us to establish sharp lower and upper asymptotic bounds for the $n$th moment of the solution.
Strong completeness of SDEs and non-explosion for RDEs with coefficients having unbounded derivatives
本文研究了具有无界导数的系数的SDEs强完备性与RDEs非爆炸性，证明了全局双连续解流的存在并展示了非爆炸性结果的尖锐性。
Variability and the existence of rough integrals with irregular coefficients
本文在分数阶微积分的粗糙路径分析框架下，利用变分性证明了在Lipschitz系数一阶偏导数为有界变差时，关于Hölder连续乘性泛函积分的存在性，并应用于Hurst指数在1/3到1/2之间的fractional Brownian motions等Gaussian过程。
Functional representation and functional calculus for controlled paths
本文建立非预期泛函演算与受控路径相容族间的联系，证明逆表示定理：相容高阶受控Taylor估计的系数族可表示为基泛函的迭代垂直导数，并应用于路径依赖粗糙微分方程的适定性。
The latest generation of volatility derivatives goes beyond variance and volatility swaps and probes our ability to price realized variance and sojourn times along bridges for the underlying stock price process. In this paper, we give an operator algebraic treatment of this problem based on Dyson expansions and moment methods and discuss applications to exotic volatility derivatives. The methods are quite flexible and allow for a specification of the underlying process which is semi-parametric or even non-parametric, including state-dependent local volatility, jumps, stochastic volatility and …
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…
The strong convergence rate of the Euler scheme for SDEs driven by additive fractional Brownian motions is studied, where the fractional Brownian motion has Hurst parameter $H\in(\frac13,\frac12)$ and the drift coefficient is not required to be bounded. The Malliavin calculus, the rough path theory and the $2$D Young integral are utilized to overcome the difficulties caused by the low regularity of the fractional Brownian motion and the unboundedness of the drift coefficient. The Euler scheme is proved to have strong order $2H$ for the case that the drift coefficient has bounded derivatives up…
Remainders of generalised Taylor expansions and a priori bounds for rough differential equations
研究由α - 赫尔德路径驱动的粗糙微分方程解的全局先验估计，利用系数及其导数特定组合的利普希茨连续性，通过广义泰勒展开余项的封闭公式来实现。
We study Runge-Kutta methods for rough differential equations which can be used to calculate solutions to stochastic differential equations driven by processes that are rougher than a Brownian motion. We use a Taylor series representation (B-series) for both the numerical scheme and the solution of the rough differential equation in order to determine conditions that guarantee the desired order of the local error for the underlying Runge-Kutta method. Subsequently, we prove the order of the global error given the local rate. In addition, we simplify the numerical approximation by introducing a…
On moments of the derivative of CUE characteristic polynomials and the Riemann zeta function
本文研究了N×N Haar分布酉矩阵特征多项式导数的复值矩，得到了显式公式，并探索其与远离临界线的黎曼zeta函数导数矩的联系，在Lindelöf假设下证明正整数矩与随机矩阵结果一致，同时提出非整数矩的渐近猜想。
In this paper we introduce the theory of derivatives of moments and (moment) functionals to represent moment functionals by Gaussian mixtures, characteristic functions of polytopes, and simple functions of polytopes. We study, among other measures, Gaussian mixtures, their reconstruction from moments and especially the number of Gaussians needed to represent moment functionals. We find that there are moment functionals $L:\mathbb{R}[x_1,\dots,x_n]_{\leq 2d}\to\mathbb{R}$ which can be represented by a sum of $\binom{n+2d}{n} - n\cdot \binom{n+d}{n} + \binom{n}{2}$ Gaussians but not less. Hence,…
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