We prove sufficient conditions, ensuring that a sequence of multiple Wiener-Itô integrals (with respect to a general Gaussian process) converges stably to a mixture of normal distributions. Our key tool is an asymptotic decomposition of contraction kernels, realized by means of increasing families of projection operators. We also use an infinite-dimensional Clark-Ocone formula, as well as a version of the correspondence between "abstract" and "concrete" filtered Wiener spaces, in a spirit similar to Üstünel and Zakai (1997).
We clearly characterize the relation between real and complex Wiener-Ito integrals. Given a complex multiple Wiener-Ito integral, we get explicit expressions for two kernels of its real and imaginary parts. Conversely, consider a two-dimensional real Wiener-Ito integral, we obtain the representation formula by a finite sum of complex Wiener-Ito integrals. The main tools are a recursion technique and Malliavin derivative operators. We build a bridge between real and complex Wiener-Ito integrals.
We present the product formula for complex multiple Wiener-Ito integrals. As applications, we show the Ustunel-Zakai independent criterion, the Nourdin-Rosinski asymptotic moment-independent criterion and joint convergence criterion for complex multiple Wiener-Ito integrals.
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 investigate the optimal rate of convergence in the multidimensional normal approximation of vector-valued Wiener-Ito integrals of which components all belong to the same fixed Wiener chaos. Combining Malliavin calculus, Stein's method for normal approximation and method of cumulants, we obtain the optimal rate of convergence with respect to a suitable smooth distance. As applications, we derive the optimal rates of convergences for complex Wiener-Ito integrals, vector-valued Wiener-Ito integrals with kernels of step functions and vector-valued Toeplitz quadratic functionals.
We define Wiener integrals with respect to Yeh processes and study their properties. In particular, we obtain the martingale property of the associated stochastic processes and give a series expansion of Wiener integrals with respect to centered Yeh process. Moreover, we derive a representation of an Yeh process in terms of a random series.
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.
For a sequence of complex Wiener-Ito multiple integrals, the equivalence between the convergence of the symmetrized contraction norms and that of the non-symmetrized contraction norms is shown directly by means of a new version of complex Mallivian calculus using the Wirtinger derivatives of complex-valued functions.
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.
Wiener integral for the coordinate process is defined under the $ σ$-finite measure unifying Brownian penalisations, which has been introduced by Najnudel, Roynette and Yor. Its decomposition before and after last exit time from 0 is studied. This study prepares for the author's recent study of Cameron--Martin formula for the $ σ$-finite measure.
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.
Some parts of stochastic analysis on curved spaces are revisted. A concise proof of the quasi-invariance of the Wiener measure on the path spaces over a Riemannian manifold is presented. The shifts are allowed to be in the Cameron-Martin space and random. The second part of the paper presents some remarks on the anticipative integrals on Riemannian manifolds.
We study the convergence to the multiple Wiener-Itô integral from processes with absolutely continuous paths. More precisely, consider a family of processes, with paths in the Cameron-Martin space, that converges weakly to a standard Brownian motion in $\mathcal C_0([0,T])$. Using these processes, we construct a family that converges weakly, in the sense of the finite dimensional distributions, to the multiple Wiener-Itô integral process of a function $f\in L^2([0,T]^n)$. We prove also the weak convergence in the space $\mathcal C_0([0,T])$ to the second order integral for two important famili…
In this article we illustrate the relation between the existence of Wiener integrals with respect to a Levy process in a separable Banach space and radonifying operators. For this purpose, we introduce the class of theta-radonifying operators, i.e. operators which map a cylindrical measure theta to a genuine Radon measure. We study this class of operators for various examples of infinitely divisible cylindrical measures theta and highlight the differences from the Gaussian case.
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…
We characterize the asymptotic independence between blocks consisting of multiple Wiener-Itô integrals. As a consequence of this characterization, we derive the celebrated fourth moment theorem of Nualart and Peccati, its multidimensional extension and other related results on the multivariate convergence of multiple Wiener-Itô integrals, that involve Gaussian and non Gaussian limits. We give applications to the study of the asymptotic behavior of functions of short and long-range dependent stationary Gaussian time series and establish the asymptotic independence for discrete non-Gaussian chao…
The two-parameter tempered Hermite field modifies the power law kernel in the moving average representation of the Hermite field by adding an exponential tempering. This paper develops the basic theory of two-parameter tempered Hermite field, including moving average, sample path properties, spectral representations and the theory of Wiener stochastic integration with respect to the two-parameter tempered Hermite field of order one.
We consider the Wiener integral with respect to a $d$-parameter Hermite process with Hurst multi-index ${\bf H}= (H_{1},\ldots, H_{d}) \in \left( \frac{1}{2}, 1\right) ^{d}$ and we analyze the limit behavior in distribution of this object when the components of ${\bf H}$ tend to $1$ and/or $\frac{1}{2}$. As examples, we focus on the solution to the stochastic heat equation with additive Hermite noise and to the Hermite Ornstein-Uhlenbeck process.
In this paper I prove good estimates on the moments and tail distribution of $k$-fold Wiener--Itô integrals and also present their natural counterpart for polynomials of independent Gaussian random variables. The proof is based on the so-called diagram formula for Wiener--Itô integrals which yields a good representation for their products as a sum of such integrals. I intend to show in a subsequent paper that this method also yields good estimates for degenerate $U$-statistics. The main result of this paper is a generalization of the estimates of Hanson and Wright about bilinear forms of indep…
We consider sequences of random variables living in a finite sum of Wiener chaoses. We find necessary and sufficient conditions for convergence in law to a target variable living in the sum of the first two Wiener chaoses. Our conditions hold notably for sequences of multiple Wiener integrals. Malliavin calculus and in particular the Gamma-operators are used. Our results extend previous findings by Azmoodeh, Peccati and Poly (2014) and are applied to central and non-central convergence situations. Our methods are applied as well to investigate stable convergence. We finally exclude certain cla…
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…
Let $\{X_{1}(t)\}_{0\leq t\leq1}$ and $\{X_{2}(t)\}_{0\leq t\leq1}$ be two independent continuous centered Gaussian processes with covariance functions$R_{1}$ and $R_{2}$. This paper shows that if the covariance functions are of finite $p$-variation and $q$-variation respectively and such that $p^{-1}+q^{-1}>1$,then the L{é}vy area can be defined as a double Wiener--Itò integral with respect to an isonormal Gaussian process induced by $X_{1}$ and $X_{2}$. Moreover, some properties of the characteristic function of that generalised L{é}vy area are studied.
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.
The problem of approximating/tracking the value of a Wiener process is considered. The discretization points are placed at times when the value of the process differs from the approximation by some amount, here denoted by eta. It is found that the limiting difference, as eta goes to 0, between the approximation and the value of the process normalized with eta converges in distribution to a triangularly distributed random variable.
The Hermite random field has been introduced as a limit of some weighted Hermite variations of the fractional Brownian sheet. In this work we define it as a multiple integral with respect to the standard Brownian sheet and introduce Wiener integrals with respect to it. As an application we study the wave equation driven by the Hermite sheet. We prove the existence of the solution and we study the regularity of its sample paths, the existence of the density and of its local times. 2000 AMS Classification Numbers: 60F05, 60H05, 60G18.
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…
We prove an extension of the Ocone-Karatzas integral representation, valid for all $BV$ functions on the classical Wiener space. We establish also an elementary chain rule formula and combine the two results to compute explicit integral representations for some classes of $BV$ composite random variables.
New representations of the Hu-Meyer formulas and series expansion of iterated Stratonovich stochastic integrals with respect to components of a multidimensional Wiener process
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…
Path integrals developed by Richard Feynman have been an important tool in Physics in studying quantum field theory. In mathematics, it has also been widely used in providing formal proofs in the study of Index theorem and asymptotic behaviors of heat kernels. Finite dimensional approximations to path integral representations give a way to interpret path integrals and make the formal argument rigorous. The central idea is to restrict a path integral to smaller path spaces where everything is well defined and then to interpret the original path integral as a "limit" when smaller path spaces "ex…
In an abstract Wiener space setting, we constract a rigorous mathematical model of the one-loop approximation of the perturbative Chern-Simons integral, and derive its explicit asymptotic expansion for stochastic Wilson lines.
We develop a stochastic analysis for a Gaussian process $X$ with singular covariance by an intrinsic procedure focusing on several examples such as covariance measure structure processes, bifractional Brownian motion, processes with stationary increments. We introduce some new spaces associated with the self-reproducing kernel space and we define the Paley-Wiener integral of first and second order even when $X$ is only a square integrable process continuous in $L^2$. If $X$ has stationary increments, we provide necessary and sufficient conditions so that its paths belong to the self-reproducin…
This paper is devoted to a construction of the stochastic Itô integral with respect to infinite dimensional cylindrical Wiener process. The construction given is an alternative one to that introduced by DaPrato and Zabczyk [3]. The connection of the introduced integral with the integral defined by Walsh [9] is provided as well.
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 study the existence of random periodic solutions for semilinear SPDEs on a bounded domain with a smooth boundary. We identify them as the solutions of coupled forward-backward infinite horizon stochastic integral equations on $L^2(D)$ in general cases. For this we use Mercer's Theorem and eigenvalues and eigenfunctions of the second order differential operators in the infinite horizon integral equations. We then use the argument of the relative compactness of Wiener-Sobolev spaces in $C^0([0, T], L^2(Ω\times D))$ and generalized Schauder's fixed point theorem to prove the exi…
In this article, we propose a way to consider processes indexed by a collection $\mathcal{A}$ of subsets of a general set $\mathcal{T}$. A large class of vector spaces, manifolds and continuous $\mathbb{R}$-trees are particular cases. Lattice-theoretic and topological assumptions are considered separately with a view to clarifying the exposition. We then define a Wiener-type integral $Y_A = \int_A f\,\text dX$ for all $A\in\mathcal{A}$ for a deterministic function $f:\mathcal{T} \rightarrow \mathbb{R}$ and a set-indexed Lévy process $X$. It is a particular case of Raput and Rosinski [40], but …
We investigate generalizations of the Cramér theorem. This theorem asserts that a Gaussian random variable can be decomposed into the sum of independent random variables if and only if they are Gaussian. We prove asymptotic counterparts of such decomposition results for multiple Wiener integrals and prove that similar results are true for the (asymptotic) decomposition of the semicircular distribution into free multiple Wigner integrals.
This paper focuses on the time-changed Q-Wiener process, a Hilbert space-valued sub-diffusion. It is a martingale with respect to an appropriate filtration, hence a stochastic integral with respect to it is definable. For the resulting integral, two change of variables formulas are derived. Via a duality theorem for integrals, existence and uniqueness theorems for stochastic differential equations (SDEs) driven by the time-changed Q-Wiener process are discussed. Associated fractional Fokker-Planck-Kolmogorov equations are derived using either a time-changed Itô formula or duality. Connections …
We prove convergence of symmetric diffusions on Wiener spaces by using stopping times arguments and capacity techniques. The drifts of the diffusions can be singular, we require the densities of the processes to be neither bounded from above nor away from zero.
Let $Z^{α,β}$ be the Rosenblatt sheet with the representation $$ Z^{α,β}(t,s)=\int^t_0\int^s_0\int^t_0\int^s_0Q^α(t,y_1,y_2)Q^β(s,u_1,u_2)B(dy_1,du_1)B(dy_2,du_2) $$ where $B$ is a Brownian sheet, $\frac12<α,β<1$, $Q^α$ and $Q^β$ are the given kernel. In this paper, we contruct multiple Wiener integrals of the form \begin{align*} \int^t_0\int^s_0\int^t_0\int^s_0&[k_1(y_1,y_2)^{-\frac12α}(u_1,u_2)^{-\frac12β}+k_2(y_1\vee y_2)^{\frac12α}(y_1\wedge y_2)^{-\frac12α}|y_1-y_2|^{α-1}\\ &\cdot(u_1\vee u_2)^{\frac12β}(u_1\wedge u_2)^{-\frac12β}|u_1-u_2|^{β-1}]B(dy_1,du_1)B(dy_2,du_2),~~k_1,k_2\geq0, \e…
Optimal Gaussian approximation for complex Wiener-Itô integrals
Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；School of Big Data, Baoshan University（保山学院大数据学院）；LMAM, School of Mathematical Sciences, Peking University（北京大学数学科学学院）
A new proof of the expansion of iterated Ito stochastic integrals with respect to the components of a multidimensional Wiener process based on generalized multiple Fourier series and Hermite polynomials
Following the ideas of F. Russo and P. Vallois we use the notion of forward integral to introduce a new stochastic integral respect to the cylindrical Winer process. This integral is an extension of the classical integral. As an application, we prove existence of solution of a parabolic stochastic differential partial equation with anticipating stochastic initial date.
Given a reference random variable, we study the solution of its Stein equation and obtain universal bounds on its first and second derivatives. We then extend the analysis of Nourdin and Peccati by bounding the Fortet-Mourier and Wasserstein distances from more general random variables such as members of the Exponential and Pearson families. Using these results, we obtain non-central limit theorems, generalizing the ideas applied to their analysis of convergence to Normal random variables. We do these in both Wiener space and the more general Wiener-Poisson space. In the former, we study condi…
In this paper by calculating carefully the capacities (defined by high order Sobolev norms on the Wiener space) for some functions of Brownian motion, we show that the dyadic approximations of the sample paths of the Brownian motion converge in the $p$-variation distance to the Brownian motion except for a slim set (i.e. except for a zero subset with respect to the capacity on the Wiener space of any order). This presents a way for studying quasi-sure properties of Wiener functionals by means of the rough path analysis.
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…
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…
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…
A parallel neighborhood of a path of a Brownian motion is sometimes called the Wiener sausage. We consider almost sure approximations of this random set by a sequence of random polyconvex sets and show that the convergence of the corresponding mean curvature measures holds under certain conditions in two and three dimensions. Based on these convergence results, the mean curvature measures of the Wiener sausage are calculated numerically by Monte Carlo simulations in two dimensions. The corresponding approximation formulae are given.
Cubature on Wiener space [Lyons, T.; Victoir, N.; Proc. R. Soc. Lond. A 8 January 2004 vol. 460 no. 2041 169-198] provides a powerful alternative to Monte Carlo simulation for the integration of certain functionals on Wiener space. More specifically, and in the language of mathematical finance, cubature allows for fast computation of European option prices in generic diffusion models. We give a random walk interpretation of cubature and similar (e.g. the Ninomiya--Victoir) weak approximation schemes. By using rough path analysis, we are able to establish weak convergence for general path-depen…
In this paper we prove that the centered three-dimensional Wiener sausage can be strongly approximated by a one-dimensional Brownian motion running at a suitable time clock. The strong approximation gives all possible laws of iterated logarithm as well as the convergence in law in terms of process for the normalized Wiener sausage. The proof relies on Le Gall's estimates between the Wiener sausage and the Brownian intersection local times.
The purpose of this paper is to provide a more general Cameron-Storvick theorem for the generalized analytic Feynman integral associated with Gaussian process $\mathcal Z_k$ on a very general Wiener space $C_{a,b}[0,T]$. The general Wiener space $C_{a,b}[0,T]$ can be considered as the set of all continuous sample paths of the generalized Brownian motion process determined by continuous functions $a(t)$ and $b(t)$ on $[0,T]$. As an interesting application, we apply this theorem to evaluate the generalized analytic Feynman integral of certain monomials in terms of Paley-Wiener-Zygmund stochastic…
