Error calculus and regularity of Poisson functionals : the lent particle method

Brownian and fractional Brownian stochastic currents via Malliavin calculus

By using Malliavin calculus and multiple Wiener-Itô integrals, we study the existence and the regularity of stochastic currents defined as Skorohod (divergence) integrals with respect to the Brownian motion and to the fractional Brownian motion. We consider also the multidimensional multiparameter case and we compare the regularity of the current as a distribution in negative Sobolev spaces with its regularity in Watanabe space.

Infinite-dimensional calculus under weak spatial regularity of the processes

Two generalizations of Itô formula to infinite-dimensional spaces are given. The first one, in Hilbert spaces, extends the classical one by taking advantage of cancellations, when they occur in examples and it is applied to the case of a group generator. The second one, based on the previous one and a limit procedure, is an Itô formula in a special class of Banach spaces, having a product structure with the noise in a Hilbertian component, again the key point is the extension due to a cancellation. This extension to Banach spaces and in particular the specific cancellation are motivated by pat…

Regularity of stochastic Volterra equations by functional calculus methods

We establish pathwise continuity properties of solutions to a stochastic Volterra equation with an additive noise term given by a local martingale. The deterministic part is governed by an operator with an $H^\infty$-calculus and a scalar kernel. The proof relies on the dilation theorem for positive definite operator families on a Hilbert space.

Malliavin Calculus and Density for Singular Stochastic Partial Differential Equations

We study Malliavin differentiability of solutions to sub-critical singular parabolic stochastic partial differential equations (SPDEs) and we prove the existence of densities for a class of singular SPDEs. Both of these results are implemented in the setting of regularity structures. For this we construct renormalized models in situations where some of the driving noises are replaced by deterministic Cameron-Martin functions, and we show Lipschitz continuity of these models with respect to the Cameron-Martin norm. In particular, in many interesting situations we obtain a convergence and stabil…

Elliptic regularity results: n-regularized Liouville Brownian motion and non-symmetric diffusions associated with degenerate forms

We apply improved elliptic regularity results to a concrete symmetric Dirichlet form and various non-symmetric Dirichlet forms with possibly degenerate symmetric diffusion matrix. Given the (non)-symmetric Dirichlet form, using elliptic regularity results and stochastic calculus we show weak existence of the corresponding singular stochastic differential equation for any starting point in some subset E of R^d. As an application of our approach we can show the existence of n-regularized Liouville Brownian motion only via Dirichlet form theory starting from all points in R^2.

正则结构中的Malliavin微积分讲义

Lecture notes on Malliavin calculus in regularity structures

本文讲义通过将Malliavin导数视为解流形切向量，结合谱间隙不等式，为次临界PDE（如随机量子化$\phi^4$模型）的模型估计提供几何化、代数化的归纳微积分框架。

Non-degeneracy of Wiener functionals arising from rough differential equations

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.

使用傅里叶分析方法研究SDE密度的正则性质

Regularity properties of densities of SDEs using the Fourier analytic approach

本文使用傅里叶分析方法研究了SDE解的局部密度的正则性质，恢复并扩展了先前基于Malliavin calculus的结论，特别是在漂移系数局部分段Hölder连续条件下推导了密度的Hölder连续性和联合连续性。

Zhang $L^2$-Regularity for the solutions of Backward Doubly Stochastic Differential Equations under globally Lipschitz continuous assumptions

We prove an $L^2$-regularity result for the solutions of Forward Backward Doubly Stochastic Differentiel Equations (FBDSDEs in short) under globally Lipschitz continuous assumptions on the coefficients. Therefore, we extend the well known regularity results established by Zhang (2004) for Forward Backward Stochastic Differential Equations (FBSDEs in short) to the doubly stochastic framework. To this end, we prove (by Malliavin calculus) a representation result for the martingale component of the solution of the F-BDSDE under the assumption that the coefficients are continuous in time and conti…

Weak quantitative propagation of chaos via differential calculus on the space of measures

Consider the metric space $(\mathcal{P}_2(\mathbb{R}^d),W_2)$ of square integrable laws on $\mathbb{R}^d$ with the topology induced by the 2-Wasserstein distance $W_2$. Let $Φ: \mathcal{P}_2( \mathbb{R}^d) \to \mathbb{R}$ be a function and $μ_N$ be the empirical measure of a sample of $N$ random variables distributed as $μ$. The main result of this paper is to show that under suitable regularity conditions, we have \[ |Φ(μ) - \mathbb{E}Φ(μ_N)|= \sum_{j=1}^{k-1}\frac{C_j}{N^j} + O(\frac{1}{N^k}), \] for some positive constants $C_1, \ldots, C_{k-1}$ that do not depend on $N$, where $k$ correspo…

Weak Functional Itô Calculus and Applications

We introduce a variational theory for processes adapted to the multi-dimensional Brownian motion filtration. The theory provides a differential structure which describes the infinitesimal evolution of Wiener functionals at very small scales. The Markov property is replaced by a finite-dimensional approximation procedure based on controlled inter-arrival times and jumps of approximating martingales. The theory reveals that a large class of adapted processes follow a differential rule which is similar in nature to a fundamental theorem of calculus in the context of Wiener functionals. Null stoch…

Statistical aspects of the fractional stochastic calculus

We apply the techniques of stochastic integration with respect to fractional Brownian motion and the theory of regularity and supremum estimation for stochastic processes to study the maximum likelihood estimator (MLE) for the drift parameter of stochastic processes satisfying stochastic equations driven by a fractional Brownian motion with any level of Hölder-regularity (any Hurst parameter). We prove existence and strong consistency of the MLE for linear and nonlinear equations. We also prove that a version of the MLE using only discrete observations is still a strongly consistent estimator.

离散随机极大 $ L^p $-正则性及线性随机热方程空间半离散化的收敛性

Discrete stochastic maximal $ L^p $-regularity and convergence of a spatial semidiscretization for a linear stochastic heat equation

本文证明了离散负 Laplace 算子的 \\( H^\infty \\)-泛函演算关于空间网格大小一致有界，并据此导出了线性随机热方程空间半离散化的离散随机极大 \\( L^p \\)-正则性估计及近乎最优的逐路径一致收敛估计。

From constructive field theory to fractional stochastic calculus. (II) Constructive proof of convergence for the Lévy area of fractional Brownian motion with Hurst index $α\in(1/8,1/4)$

{Let $B=(B_1(t),...,B_d(t))$ be a $d$-dimensional fractional Brownian motion with Hurst index $α<1/4$, or more generally a Gaussian process whose paths have the same local regularity. Defining properly iterated integrals of $B$ is a difficult task because of the low Hölder regularity index of its paths. Yet rough path theory shows it is the key to the construction of a stochastic calculus with respect to $B$, or to solving differential equations driven by $B$. We intend to show in a series of papers how to desingularize iterated integrals by a weak, singular non-Gaussian perturbation of the Ga…

基于分数阶微积分技术的分数布朗运动驱动随机泛函微分方程的欧拉格式

Euler Scheme for Stochastic Functional Differential Equations Driven by Fractional Brownian Motion via Fractional Calculus Techniques

针对Hurst参数H>1/2的分数布朗运动驱动的带记忆随机泛函微分方程，提出并分析了基于分数阶微积分的欧拉型数值格式，证明了收敛性并给出了收敛阶。