S. V. Lototsky, B. L. Rozovskii
A new method is described for constructing a generalized solution of a stochastic evolution equation. Existence, uniqueness, regularity and a probabilistic representation of this Wiener Chaos solution are established for a large class of equations. As an application of the general theory, new results are obtained for several types of the passive scalar equation.
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In this article, some properties of complex Wiener-Itô multiple integrals and complex Ornstein-Uhlenbeck operators and semigroups are obtained. Those include Stroock's formula, Hu-Meyer formula, Clark-Ocone formula and the hypercontractivity of complex Ornstein-Uhlenbeck semigroups. As an application, several expansions of the fourth moments of complex Wiener-Itô multiple integrals are given.
Infinite divisibility of a class of two-dimensional vectors with components in the second Wiener chaos is studied. Necessary and sufficient conditions for infinite divisibility is presented as well as more easily verifiable sufficient conditions. The case where both components consist of a sum of two Gaussian squares is treated in more depth, and it is conjectured that such vectors are infinitely divisible.
We present an algorithm to solve BSDEs with jumps based on Wiener Chaos Expansion and Picard's iterations. This paper extends the results given in Briand-Labart (2014) to the case of BSDEs with jumps. We get a forward scheme where the conditional expectations are easily computed thanks to chaos decomposition formulas. Concerning the error, we derive explicit bounds with respect to the number of chaos, the discretization time step and the number of Monte Carlo simulations. We also present numerical experiments. We obtain very encouraging results in terms of speed and accuracy.
We compute explicit bounds in the normal and chi-square approximations of multilinear homogenous sums (of arbitrary order) of general centered independent random variables with unit variance. In particular, we show that chaotic random variables enjoy the following form of universality: (a) the normal and chi-square approximations of any homogenous sum can be completely characterized and assessed by first switching to its Wiener chaos counterpart, and (b) the simple upper bounds and convergence criteria available on the Wiener chaos extend almost verbatim to the class of homogeneous sums.
We combine Malliavin calculus with Stein's method, in order to derive explicit bounds in the Gaussian and Gamma approximations of random variables in a fixed Wiener chaos of a general Gaussian process. We also prove results concerning random variables admitting a possibly infinite Wiener chaotic decomposition. Our approach generalizes, refines and unifies the central and non-central limit theorems for multiple Wiener-Itô integrals recently proved (in several papers, from 2005 to 2007) by Nourdin, Nualart, Ortiz-Latorre, Peccati and Tudor. We apply our techniques to prove Berry-Esséen bounds in…
A new method is described for constructing a generalized solution for stochastic differential equations. The method is based on the Cameron-Martin version of the Wiener Chaos expansion and provides a unified framework for the study of ordinary and partial differential equations driven by finite- or infinite-dimensional noise with either adapted or anticipating input. Existence, uniqueness, regularity, and probabilistic representation of this Wiener Chaos solution is established for a large class of equations. A number of examples are presented to illustrate the general constructions. A detaile…
We present an algorithm to solve BSDEs based on Wiener chaos expansion and Picard's iterations. We get a forward scheme where the conditional expectations are easily computed thanks to chaos decomposition formulas. We use the Malliavin derivative to compute $Z$. Concerning the error, we derive explicit bounds with respect to the number of chaos and the discretization time step. We also present numerical experiments. We obtain very encouraging results in terms of speed and accuracy.
Let $F_n = (F_{1,n}, ....,F_{d,n})$, $n\geq 1$, be a sequence of random vectors such that, for every $j=1,...,d$, the random variable $F_{j,n}$ belongs to a fixed Wiener chaos of a Gaussian field. We show that, as $n\to\infty$, the components of $F_n$ are asymptotically independent if and only if ${\rm Cov}(F_{i,n}^2,F_{j,n}^2)\to 0$ for every $i\neq j$. Our findings are based on a novel inequality for vectors of multiple Wiener-Itô integrals, and represent a substantial refining of criteria for asymptotic independence in the sense of moments recently established by Nourdin and Rosinski.
Let L be the class of limiting laws associated with sequences in the second Wiener chaos. We exhibit a large subset L_0 of L satisfying that, for any F_infinity in L_0, the convergence of only a finite number of cumulants suffices to imply the convergence in law of any sequence in the second Wiener chaos to F_infinity. This result is in the spirit of the seminal paper by Nualart and Peccati, in which the authors discovered the surprising fact that convergence in law for sequences of multiple Wiener-Itô integrals to the Gaussian is equivalent to convergence of just the fourth cumulant. Also, we…
A moderate deviation principle as well as moderate and large deviation inequalities for a sequence of elements living inside a fixed Wiener chaos associated with an isonormal Gaussian process are shown. The conditions under which the results are derived coincide with those of the celebrated fourth moment theorem of Nualart and Peccati. The proofs rely on sharp estimates for cumulants. As applications, explosive integrals of a Brownian sheet, a discretized version of the quadratic variation of a fractional Brownian motion and the sample bispectrum of a spherical Gaussian random field are consid…
In [NP09a], Nourdin and Peccati established a neat characterization of Gamma approximation on a fixed Wiener chaos in terms of convergence of only the third and fourth cumulants. In this paper, we investigate the rate of convergence in Gamma approximation on Wiener chaos in terms of the iterated Gamma operators of Malliavin Calculus. On the second Wiener chaos, our upper bound can be further extended to an exact rate of convergence in a suitable probability metric $d_2$ in terms of the maximum of the third and fourth cumulants, analogous to that of normal approximation in [NP15] under one extr…
Fractional Wiener Chaos
该研究将分数阶微积分引入随机分析，基于带指数因子的幂归一化抛物柱函数构建分数阶维纳混沌展开，证明其为埃尔米特多项式的分数阶推广，保留鞅等核心性质，拓展了维纳混沌理论的分数阶研究方向。
In this survey we collect some recent results regarding the Lipschitz-Killing curvatures (LKCs) of the excursion sets of random eigenfunctions on the two-dimensional standard flat torus (arithmetic random waves) and on the two-dimensional unit sphere (random spherical harmonics). In particular, the aim of the present survey is to highlight the key role of integration by parts formulae in order to have an extremely neat expression for the random LKCs. Indeed, the main tool to study local geometric functionals of random waves on manifold is to exploit their Wiener chaos decomposition and show th…
We develop the asymptotic expansion theory for vector-valued sequences (F N) N $\ge$1 of random variables in terms of the convergence of the Stein-Malliavin matrix associated to the sequence F N. Our approach combines the classical Fourier approach and the recent theory on Stein method and Malliavin calculus. We find the second order term of the asymptotic expansion of the density of F N and we illustrate our results by several examples. 2010 AMS Classification Numbers: 62M09, 60F05, 62H12
In \cite{BNT}, a framework to prove almost sure central limit theorems for sequences $(G_n)$ belonging to the Wiener space was developed, with a particular emphasis of the case where $G_n$ takes the form of a multiple Wiener-Itô integral with respect to a given isonormal Gaussian process. In the present paper, we complement the study initiated in \cite{BNT}, by considering the more general situation where the sequence $(G_n)$ may not need to converge to a Gaussian distribution. As an application, we prove that partial sums of Hermite polynomials of increments of fractional Brownian motion sati…
The chaos expansion of a general non-linear function of a Gaussian stationary increment process conditioned on its past realizations is derived. This work combines Wiener chaos expansion approach to study the dynamics of a stochastic system with the classical problem of the prediction of a Gaussian process based on a realization of its past. This is done by constructing a special basis for the Fock space of the Gaussian space generated by the process, such that each basis element is either measurable or independent with respect to the given samples. This property of the basis allows us to deri…
In the first part of the paper we use a new Fourier technique to obtain a Stein characterizations for random variables in the second Wiener chaos. We provide the connection between this result and similar conclusions that can be derived using Malliavin calculus. We also introduce a new form of discrepancy which we use, in the second part of the paper, to provide bounds on the 2-Wasserstein distance between linear combinations of independent centered random variables. Our method of proof is entirely original. In particular it does not rely on estimation of bounds on solutions of the so-called S…
Let $Z$ be a standard normal random variable and let $H_n$ denote the $n$-th Hermite polynomial. In this note, we obtain Stein equations for the random variables $H_3(Z)$ and $H_4(Z)$, which represents a first step towards developing Stein's method for distributional limits from the third and fourth Wiener chaoses. Perhaps surprisingly, these Stein equations are fifth and third order linear ordinary differential equations, respectively. As a warm up, we obtain a Stein equation for the random variable $aZ^2+bZ+c$, $a,b,c\in\mathbb{R}$, which leads us to a Stein equation for the non-central chi-…
Optimal local central limit theorems on Wiener chaos
本文在韦纳混沌中研究了随机变量序列的局部中心极限定理，证明了密度函数收敛速率由第三和第四累积量的最大绝对值决定。
On the Wiener Chaos Expansion of the Signature of a Gaussian Process
该研究推导了含H∈(1/4,1)的分数布朗运动在内的一类高斯过程签名的维纳混沌分解，得到其期望签名表达式，将已有相关公式推广到H>1/4的一般情形，解决了公开问题，还研究了连续中心化高斯半鞅。
We consider Gaussian random waves on hyperbolic spaces and establish variance asymptotics and central limit theorems for a large class of their integral functionals, both in the high-frequency and large domain limits. Our strategy of proof relies on a fine analysis of Wiener chaos expansions, which in turn requires us to analytically assess the fluctuations of integrals involving mixed moments of covariance kernels. Our results complement several recent findings on non-linear transforms of planar and arithmetic random waves, as well as of random spherical harmonics. In the particular case of 2…
Simulating diffusion bridges using the Wiener chaos expansion
本文提出基于Wiener混沌展开的扩散桥模拟方法，速度快且无拒绝率，通过Ornstein-Uhlenbeck过程验证，并应用于三个SDE示例，适用于统计推断。
In this paper, a product formula of Hermite polynomials is given and then the relation between the real Wiener-Itô chaos and the complex Wiener-Itô chaos (or: multiple integrals) is shown. By this relation and the known multivariate extension of the fourth moment theorem for the real multiple integrals, the fourth moment theorem (or say: the Nualart-Peccati criterion) for the complex Wiener-Itô multiple integrals is obtained.
The aim of this paper is to establish some new results on the absolute continuity and the convergence in total variation for a sequence of d-dimensional vectors whose components belong to a finite sum of Wiener chaoses. First we show that the probability that the determinant of the Malliavin matrix of such vectors vanishes is zero or one, and this probability equals to one is equivalent to say that the vector takes values in the set of zeros of a polynomial. We provide a bound for the degree of this annihilating polynomial improving a result by Kusuoka. On the other hand, we show that the conv…
In this paper, we consider a target random variable $Y \sim \CVG$ distributed according to a centered Variance--Gamma distribution. For a generic random element $F=I_2(f)$ in the second Wiener chaos with $\E[F^2]= \E[Y^2]$ we establish a non-asymptotic optimal bound on the distance between $F$ and $Y$ in terms of the maximum of difference of the first six cumulants. This six moment theorem extends the celebrated optimal fourth moment theorem of I.\ Nourdin \& G.\ Peccati for normal approximation. The main body of our analysis constitutes a splitting technique for test functions in the Banach s…
Consider $F$ an element of the second Wiener chaos with variance one. In full generality, we show that, for every integer $p\ge 1$, there exists $η_p>0$ such that if $κ_4(F)<η_p$ then the Malliavin derivative of $F$ admits a negative moment of order $p$. This entails that any sequence of random variables in the second Wiener chaos converging in distribution to a non--degenerated Gaussian is getting more regular as its distribution is getting close to the normal law. This substantially generalizes some recent findings contained in \cite{hu2014convergence,hu2015density,nourdin2016fisher} where a…
Let ${F_n}$ be a sequence of random variables belonging to a finite sum of Wiener chaoses. Assume further that it converges in distribution towards $F_\infty$ satisfying ${\rm Var}(F_\infty)>0$. Our first result is a sequential version of a theorem by Shigekawa (1980). More precisely, we prove, without additional assumptions, that the sequence ${F_n}$ actually converges in total variation and that the law of $F_\infty$ is absolutely continuous. We give an application to discrete non-Gaussian chaoses. In a second part, we assume that each $F_n$ has more specifically the form of a multiple Wiene…
We prove that the formula which gives the Wiener chaos decomposition of the multiplication of two multiple Wiener integrals with symmetric kernels is a straightforward application of the Leibniz' formula.
We consider a process $\{X_t\}_{0\leq t\leq 1}$ in a fixed Wiener chaos $\mathcal{H}_n$. We establish some non-degenerate properties and related results for $\{X_t\}_{0\leq t\leq 1}$. As an application, we show that solution to SDE driven by $\{X_t\}_{0\leq t\leq 1}$ admits a density. Our approach relies on an interplay between Malliavin calculus and analysis on Wiener space.
In [14], Nourdin and Peccati combined the Malliavin calculus and Stein's method of normal approximation to associate a rate of convergence to the celebrated fourth moment theorem [19] of Nualart and Peccati. Their analysis, known as the Malliavin-Stein method nowadays, has found many applications towards stochastic geometry, statistical physics and zeros of random polynomials, to name a few. In this article, we further explore the relation between these two fields of mathematics. In particular, we construct exchangeable pairs of Brownian motions and we discover a natural link between Malliavin…
Using the structure of the Boson-Fermion Fock space and an argument taken from [2], we give a new proof of the triviality of the $L^2$ cohomology groups on an abstract Wiener space, alternative to that given by Shigekawa [9]. We apply some representation theory of the symmetric group to characterise the spaces of exact and co-exact forms in their Boson-Fermion Fock space representation.
Empirical limit theorems for Wiener chaos
本文在三角阵列框架下研究底层分布随样本量变化的经验测度，建立了归一化经验测度多重积分及多重Wiener-Itô积分级数的极限定理。
Edgeworth expansion on Wiener chaos
本文在Wiener混沌上建立了任意阶的Edgeworth展开，以carré-du-champ算子的方差为小参数，给出显式余项界，并恢复最优四阶矩定理。
In this paper, we study the distribution of the so-called "Yule's nonsense correlation statistic" on a time interval $[0,T]$ for a time horizon $T>0$ , when $T$ is large, for a pair $(X_{1},X_{2})$ of independent Ornstein-Uhlenbeck processes. This statistic is by definition equal to : \begin{equation*} ρ(T):=\frac{Y_{12}(T)}{\sqrt{Y_{11}(T)}\sqrt{Y_{22}(T)}}, \end{equation*} where the random variables $Y_{ij}(T)$, $i,j=1,2$ are defined as \begin{equation*} Y_{ij}(T):=\int_{0}^{T}X_{i}(u)X_{j}(u)du-T\bar{X}_{i}\bar{X_{j}}, \bar{X}_{i}:=\frac{1}{T}\int_{0}^{T}X_{i}(u)du. \end{equation*} We assum…
In this we paper we recast the Cox--Ingersoll--Ross model of interest rates into the chaotic representation recently introduced by Hughston and Rafailidis. Beginning with the ``squared Gaussian representation'' of the CIR model, we find a simple expression for the fundamental random variable X. By use of techniques from the theory of infinite dimensional Gaussian integration, we derive an explicit formula for the n-th term of the Wiener chaos expansion of the CIR model, for n=0,1,2,.... We then derive a new expression for the price of a zero coupon bond which reveals a connection between Gauss…
We consider a {\it{Gaussian multiplicative chaos}} (GMC) measure on the classical Wiener space driven by a smoothened (Gaussian) space-time white noise. For $d\geq 3$ it was shown in \cite{MSZ16} that for small noise intensity, the total mass of the GMC converges to a strictly positive random variable, while larger disorder strength (i.e., low temperature) forces the total mass to lose uniform integrability, eventually producing a vanishing limit. Inspired by strong localization phenomena for log-correlated Gaussian fields and Gaussian multiplicative chaos in the finite dimensional Euclidean s…
The integration-by-parts formula discovered by Malliavin for the Ito map on Wiener space is proved using the two-parameter stochastic calculus. It is also shown that the solution of a one-parameter stochastic differential equation driven by a two-parameter semimartingale is itself a two-parameter semimartingale.
In this paper, we propose a general means of estimating the rate at which convergences in law occur. Our approach, which is an extension of the classical Stein-Tikhomirov method, rests on a new pair of linear operators acting on characteristic functions. In principle, this method is admissible for any approximating sequence and any target, although obviously the conjunction of several favorable factors is necessary in order for the resulting bounds to be of interest. As we briefly discuss, our approach is particularly promising whenever some version of Stein's method applies. We apply our appr…
In this work, we propose a new policy iteration algorithm for pricing Bermudan options when the payoff process cannot be written as a function of a lifted Markov process. Our approach is based on a modification of the well-known Longstaff Schwartz algorithm, in which we basically replace the standard least square regression by a Wiener chaos expansion. Not only does it allow us to deal with a non Markovian setting, but it also breaks the bottleneck induced by the least square regression as the coefficients of the chaos expansion are given by scalar products on the L^2 space and can therefore b…
By using stochastic calculus for two-parameter processes and chaos expansion into multiple Wiener-Itô integrals, we define a 2D-stochastic current over the Brownian sheet. This concept comes from geometric measure theory. We also study the regularity of the stochastic current with respect to the randomness variable in the Watanabe spaces and with respect to the spatial variable in the deterministic Sobolev spaces.
Convergence Rate Analysis in Limit Theorems for Nonlinear Functionals of the Second Wiener Chaos
本文研究高斯过程平方包络小波变换幅值随机向量的大尺度极限，针对Hurst指数小于1/2和大于1/2两种情形，分别用非Stein方法与Malliavin分析结合多维Stein方法给出Kolmogorov和Wasserstein度量下的收敛速度，并揭示不同Hurst子区间上界的差异。
In this paper, we study the asymptotic behavior of the quadratic variation for the class of AR(1) processes driven by white noise in the second Wiener chaos. Using tools from the analysis on Wiener space, we give an upper bound for the total-variation speed of convergence to the normal law, which we apply to study the estimation of the model's mean-reversion. Simulations are performed to illustrate the theoretical results.
On the Analysis of a Singular Stochastic Volterra Differential Equation driven by a Wiener Noise
本文研究了由奇异漂移向量场和Wiener噪声驱动的随机Volterra微分方程的强解唯一性及初始值对强解的Sobolev可微性。
We consider a mixed stochastic differential equation driven by possibly dependent fractional Brownian motion and Brownian motion. Under mild regularity assumptions on the coefficients, it is proved that the equation has a unique solution.
Wiener chaos vs stochastic collocation methods for linear advection-diffusion equations with multiplicative white noise
本文比较了维纳混沌与随机格点方法在具有乘性白噪声的线性对流扩散方程中的应用，分析了两种方法的误差估计及数值性能，发现递归多阶段随机格点方法在二阶矩中具有Δ阶精度，而递归多阶段维纳混沌方法在对流扩散反应方程中具有Δ^N+Δ^2阶精度，但非交换噪声下两者均具有一阶精度。
Optimal Sobolev Rate for Gaussian Density Approximation of Wiener Chaos Vectors
本文证明Wiener混沌向量高斯逼近在Sobolev空间中的最优密度收敛速率由三阶和四阶累积量决定，并推广至多种距离，方法无需Malliavin非退化条件。
The first-passage area of Wiener process with stochastic resetting
研究带随机重置的一维维纳过程从正初始位置出发的首达时间与首达面积的统计性质，利用常微分方程解导出其拉普拉斯变换及单矩和联合矩的显式表达式。
An Euler scheme for BSDEs via the Wiener chaos decomposition
提出基于Wiener混沌分解的Euler格式实现BSDE数值求解，无需前向-后向马尔可夫结构，适用于任意平方可积终端条件，并给出收敛性分析与数值验证。
New representations of the Hu-Meyer formulas and series expansion of iterated Stratonovich stochastic integrals with respect to components of a multidimensional Wiener process
本文系统推导了Hu-Meyer公式的新型表示，针对多维维纳过程，推导了多重维纳随机积分通过多重斯特劳尼奇随机积分之和及多重斯特劳尼奇随机积分通过多重维纳随机积分之和的公式，探讨了不同斯特劳尼奇多重随机积分定义及Hu-Meyer公式的充分条件变体，为构造高阶强数值方法提供支持。
Coupling of forward-backward stochastic differential equations on the Wiener space, and application on regularity
本文将Geiss和Ylinen提出的耦合方法拓展到带随机Lipschitz系数的正倒向随机微分方程场景，得到了解的时间正则性与Malliavin Sobolev ${\u211D}_{1,2}$可微性结果。
Solving stochastic partial differential equations using neural networks in the Wiener chaos expansion
本文提出利用神经网络在维纳混沌展开中求解随机偏微分方程，并通过数值实验验证了方法的可行性与有效性。
In this work cylindrical Wiener processes on Banach spaces are defined by means of cylindrical stochastic processes, which are a well considered mathematical object. This approach allows a definition which is a simple straightforward extension of the real-valued situation. We apply this definition to introduce a stochastic integral with respect to cylindrical Wiener processes. Again, this definition is a straightforward extension of the real-valued situation which results now in simple conditions on the integrand. In particular, we do not have to put any geometric constraints on the Banach spa…
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.
We introduce the notion of {\em covariance measure structure} for square integrable stochastic processes. We define Wiener integral, we develop a suitable formalism for stochastic calculus of variations and we make Gaussian assumptions only when necessary. Our main examples are finite quadratric variation processes with stationary increments and the bifractional Brownian motion.
We develop an approach for investigating geometric properties of Gaussian multiplicative chaos (GMC) in an infinite dimensional set up. The base space is chosen to be the space of continuous functions endowed with Wiener measure, and the random field is a space-time white noise integrated against Brownian paths. In this set up, we show that in any dimension $d\geq 1$ and for any inverse temperature, the GMC-volume of a ball, uniformly around all paths, decays exponentially with an explicit decay rate. The exponential rate reflects the balance between two competing terms, namely the principal e…
An elementary construction of the Wiener process is discussed, based on a proper sequence of simple symmetric random walks that uniformly converge on bounded intervals, with probability 1. This method is a simplification of F.B. Knight's and P. Révész's. The same sequence is applied to give elementary (Lebesgue-type) definitions of Itô and Stratonovich sense stochastic integrals and to prove the basic Itô formula. The resulting approximating sums converge with probability 1. As a by-product, new elementary proofs are given for some properties of the Wiener process, like the almost sure non-dif…
Topics in Gaussian Wiener chaos expansion
本文介绍了有限维下的Wiener chaos分解，构造了在环面上的高斯场，包括白噪声和高斯自由场，并将其应用于Φ^4模型，但未涵盖随机积分、随机偏微分方程和马尔可夫算子等其他重要方面。
The solution of a (stochastic) differential equation (SDE) can be locally approximated by a stochastic expansion, a linear combination of iterated integrals. Quantities of interest, like moments, can then be approximated with the expansion. We present a formula for the case where the drivers of the equation are time and Wiener processes. We also present a Mathematica implementation of the result.
We consider a one-dimensional stochastic differential equation driven by a Wiener process, where the diffusion coefficient depends on an ergodic fast process. The averaging principle is satisfied: it is well-known that the slow component converges in distribution to the solution of an averaged equation, with generator determined by averaging the square of the diffusion coefficient. We propose a version of the averaging principle, where the solution is interpreted as the sum of two terms: one depending on the average of the diffusion coefficient, the other giving fluctuations around that averag…
Mean-variance dynamic portfolio allocation with transaction costs: a Wiener chaos expansion approach
本文提出基于维纳混沌展开的数值方法，解决带交易成本的多期平均方差投资组合配置问题，并构建单期Markowitz基准验证其效率。
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