Following a strategy recently developed by Ivan Nourdin and Giovanni Peccati, we provide a general technique to compare the tail of a given random variable to that of a reference distribution. This enables us to give concrete conditions to ensure upper and/or lower bounds on the random variable's tail of various power or exponential types. The Nourdin-Peccati strategy analyzes the relation between Stein's method and the Malliavin calculus, and is adapted to dealing with comparisons to the Gaussian law. By studying the behavior of the solution to general Stein equations in detail, we show that the strategy can be extended to comparisons to a wide class of laws, including many Pearson distributions.
The Malliavin derivative, divergence operator, and the Ornstein-Uhlenbeck operator are extended from the traditional Gaussian setting to generalized processes from the higher-order chaos spaces.
Initiated around the year 2007, the Malliavin-Stein approach to probabilistic approximations combines Stein's method with infinite-dimensional integration by parts formulae based on the use of Malliavin-type operators. In the last decade, Malliavin-Stein techniques have allowed researchers to establish new quantitative limit theorems in a variety of domains of theoretical and applied stochastic analysis. The aim of this survey is to illustrate some of the latest developments of the Malliavin-Stein method, with specific emphasis on extensions and generalisations in the framework of Markov semig…
In this paper, we give an upper bound for a probabilistic distance between a Gaussian vector and a vector of U-statistics of Poisson point processes by applying Malliavin-Stein inequality on the Poisson space.
We develop a theory of Malliavin calculus for Banach space valued random variables. Using radonifying operators instead of symmetric tensor products we extend the Wiener-Ito isometry to Banach spaces. In the white noise case we obtain two sided L^p-estimates for multiple stochastic integrals in arbitrary Banach spaces. It is shown that the Malliavin derivative is bounded on vector-valued Wiener-Ito chaoses. Our main tools are decoupling inequalities for vector-valued random variables. In the opposite direction we use Meyer's inequalities to give a new proof of a decoupling result for Gaussian …
The fourth moment theorem provides error bounds of the order $\sqrt{{\mathbb E}(F^4) - 3}$ in the central limit theorem for elements $F$ of Wiener chaos of any order such that ${\mathbb E}(F^2) = 1$. It was proved by Nourdin and Peccati (2009) using Stein's method and the Malliavin calculus. It was also proved by Azmoodeh, Campese and Poly (2014) using Stein's method and Dirichlet forms. This paper is an exposition on the connections between Stein's method and the Malliavin calculus and between Stein's method and Dirichlet forms, and on how these connections are exploited in proving the fourth…
The Stein's method is a popular method used to derive upper-bounds of distances between probability distributions. It can be viewed, in certain of its formulations, as an avatar of the semi-group or of the smart-path method used commonly in Gaussian analysis. We show how this procedure can be enriched by Malliavin calculus leading to a functional approach valid in infinite dimensional spaces.
In this paper we study the Malliavin derivatives and Skorohod integrals for processes taking values in an infinite dimensional space. Such results are motivated by their applications to SPDEs and in particular financial mathematics. Vector-valued Malliavin theory in Banach space E is naturally restricted to spaces E which have the so-called UMD property, which arises in harmonic analysis and stochastic integration theory. We provide several new results and tools for the Malliavin derivatives and Skorohod integrals in an infinite dimensional setting. In particular, we prove weak characterizatio…
In this paper, following Nourdin-Peccati's methodology, we combine the Malliavin calculus and Stein's method to provide general bounds on the Wasserstein distance between functionals of a compound Hawkes process and a given Gaussian density. To achieve this, we rely on the Poisson embedding representation of an Hawkes process to provide a Malliavin calculus for the Hawkes processes, and more generally for compound Hawkes processes. As an application, we close a gap in the literature by providing the first Berry-Esséen bounds associated to Central Limit Theorems for the compound Hawkes process.
Given a random variable $F$ regular enough in the sense of the Malliavin calculus, we are able to measure the distance between its law and almost any continuous probability law on the real line. The bounds are given in terms of the Malliavin derivative of $F$. Our approach is based on the theory of Itô diffusions and the stochastic calculus of variations. Several examples are considered in order to illustrate our general results.
In this paper, we provide upper bounds on the d2 distance between a large class of functionals of a multivariate compound Hawkes process and a given Gaussian vector. This is proven using Malliavin's calculus defined on an underlying Poisson embedding. The upper bound is then used to infer the speed of convergence of Central Limit Theorems for the multivariate compound Hawkes process with exponential kernels as the observation time T goes to infinity.
We show how to use the Malliavin calculus to obtain density estimates of the law of general centered random variables. In particular, under a non-degeneracy condition, we prove and use a new formula for the density of a random variable which is measurable and differentiable with respect to a given isonormal Gaussian process. Among other results, we apply our techniques to bound the density of the maximum of a general Gaussian process from above and below; several new results ensue, including improvements on the so-called Borell-Sudakov inequality. We then explain what can be done when one is o…
We combine Stein's method with Malliavin calculus in order to obtain explicit bounds in the multidimensional normal approximation (in the Wasserstein distance) of functionals of Gaussian fields. Our results generalize and refine the main findings by Peccati and Tudor (2005), Nualart and Ortiz-Latorre (2007), Peccati (2007) and Nourdin and Peccati (2007b, 2008); in particular, they apply to approximations by means of Gaussian vectors with an arbitrary, positive definite covariance matrix. Among several examples, we provide an application to a functional version of the Breuer-Major CLT for field…
We compute Greeks for stochastic volatility models driven by Brownian informations. We use the Malliavin method introduced for deterministic volatility models.
New bounds on the total variation distance between the law of integer valued functionals of possibly non-symmetric and non-homogeneous infinite Rademacher sequences and the Poisson distribution are established. They are based on a combination of the Chen-Stein method and a discrete version of Malliavin calculus. We give some applications to shifted discrete multiple stochastic integrals.
We combine Malliavin calculus with Stein's method to derive bounds for the Variance-Gamma approximation of functionals of isonormal Gaussian processes, in particular of random variables living inside a fixed Wiener chaos induced by such a process. The bounds are presented in terms of Malliavin operators and norms of contractions. We show that a sequence of distributions of random variables in the second Wiener chaos converges to a Variance-Gamma distribution if and only if their moments of order two to six converge to that of a Variance-Gamma distributed random variable (six moment theorem). M…
We provide an overview of some recent techniques involving the Malliavin calculus of variations and the so-called ``Stein's method'' for the Gaussian approximations of probability distributions. Special attention is devoted to establishing explicit connections with the classic method of moments: in particular, we use interpolation techniques in order to deduce some new estimates for the moments of random variables belonging to a fixed Wiener chaos. As an illustration, a class of central limit theorems associated with the quadratic variation of a fractional Brownian motion is studied in detail.
A new Berry-Esseen bound for non-linear functionals of non-symmetric and non-homogeneous infinite Rademacher sequences is established. It is based on a discrete version of the Malliavin-Stein method and an analysis of the discrete Ornstein-Uhlenbeck semigroup. The result is applied to sub-graph counts and to the number of vertices having a prescribed degree in the Erdős-Renyi random graph. A further application deals with a percolation problem on trees.
Stein's method is a method of probability approximation which hinges on the solution of a functional equation. For normal approximation the functional equation is a first order differential equation. Malliavin calculus is an infinite-dimensional differential calculus whose operators act on functionals of general Gaussian processes. Nourdin and Peccati (2009) established a fundamental connection between Stein's method for normal approximation and Malliavin calculus through integration by parts. This connection is exploited to obtain error bounds in total variation in central limit theorems for …
We study the self-normalized sums of independent random variables from the perspective of the Malliavin calculus. We give the chaotic expansion for them and we prove a Berry-Esséen bound with respect to several distances.
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.
We give a review of our recent works related to the Malliavin Calculus of Bismut type for non-Markovian generators. Part IV is new and relates the Malliavin Calculus and the general theory of elliptic pseudo-differential operators.
In this paper, based on a known formula, we use a simple idea to get a new representation for the density of Malliavin differentiable random variables. This new representation is particularly useful for finding lower bounds for the density.
We give new contributions on the distribution of the zeros of the Riemann zeta function by using the techniques of the Malliavin calculus. In particular, we obtain the error bound in the multidimensional Selberg' s central limit theorem concerning the zeta zeros on the critical line and we discuss some consequences concerning the asymptotic behavior of the mesoscopic fluctuations of the zeta zeros.
In this paper, we construct a Malliavin derivative for functionals of square-integrable Lévy processes and derive a Clark-Ocone formula. The Malliavin derivative is defined via chaos expansions involving stochastic integrals with respect to Brownian motion and Poisson random measure. As an illustration, we compute the explicit martingale representation for the maximum of a Lévy process.
In this paper, we use Malliavin calculus to show the existence and continuity of density functions of $d$-dimensional non-colliding particle systems such as hyperbolic particle systems and Dyson Brownian motion with smooth drift. For this purpose, we apply results proved by Florit and Nualart (1995) and Naganuma (2013) on locally non-degenerate Wiener functionals.
The starting point of the current paper is a sequence of uncorrelated random variables. The distribution functions of these variables are assumed to be given but no assumptions on the types or the structure of these distributions are made. The above setting constitute the so called "distribution free" paradigm. Under these assumptions, a version of Skorokhod-Malliavin calculus is developed and applications to stochastic PDES are discussed.
By means of the Malliavin calculus, integral representations for the likelihood function and for the derivative of the log-likelihood function are given for a model based on discrete time observations of the solution to equation dX_t=a_θ(X_t)dt + dZ_t with a tempered α-stable process Z. Using these representations, regularity of the statistical experiment and the Cramer-Rao inequality are proved.
We introduce a Multifractal Random Walk (MRW) defined as a stochastic integral of an infinitely divisible noise with respect to a dependent fractional Brownian motion. Using the techniques of the Malliavin calculus, we study the existence of this object and its properties. We then propose a continuous time model in finance that captures the main properties observed in the empirical data, including the leverage effect. We illustrate our result by numerical simulations.
Suppose that $\{u(t\,, x)\}_{t >0, x \in\mathbb{R}^d}$ is the solution to a $d$-dimensional stochastic heat equation driven by a Gaussian noise that is white in time and has a spatially homogeneous covariance that satisfies Dalang's condition. The purpose of this paper is to establish quantitative central limit theorems for spatial averages of the form $N^{-d} \int_{[0,N]^d} g(u(t\,,x))\, \mathrm{d} x$, as $N\rightarrow\infty$, where $g$ is a Lipschitz-continuous function or belongs to a class of locally-Lipschitz functions, using a combination of the Malliavin calculus and Stein's method for …
In this paper a Malliavin calculus for Lévy processes based on a family of true derivative operators is developed. The starting point is an extension to Lévy processes of the pioneering paper by Carlen and Pardoux [8] for the Poisson process, and our approach includes also the classical Malliavin derivative for Gaussian processes. We obtain a sufficient condition for the absolute continuity of functionals of the Lévy process. As an application, we analyze the absolute continuity of the law of the solution of some stochastic differential equations.
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 give a new characterization for the convergence in distribution to a standard normal law of a sequence of multiple stochastic integrals of a fixed order with variance one, in terms of the Malliavin derivatives of the sequence. We extend our result to the multidimensional case and prove a weak convergence result for a sequence of square integrable random variables.
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.
In this paper we develop a Malliavin-Skorohod type calculus for additive processes in the $L^0$ and $L^1$ settings, extending the probabilistic interpretation of the Malliavin-Skorohod operators to this context. We prove calculus rules and obtain a generalization of the Clark-Hausmann-Ocone formula for random variables in $L^1$. Our theory is then applied to extend the stochastic integration with respect to volatility modulated Lévy-driven Volterra processes recently introduced in the literature. Our work yields to substantially weaker conditions that permit to cover integration with respect, …
For a one-dimensional smooth vector field in a neighborhood of an unstable equilibrium, we consider the associated dynamics perturbed by small noise. Using Malliavin calculus tools, we obtain precise vanishing noise asymptotics for the tail of the exit time and for the exit distribution conditioned on atypically long exits.
In this paper we study the existence of a unique solution to a general class of Young delay differential equations driven by a Hölder continuous function with parameter greater that 1/2 via the Young integration setting. Then some estimates of the solution are obtained, which allow to show that the solution of a delay differential equation driven by a fractional Brownian motion (fBm) with Hurst parameter H>1/2 has a smooth density. To this purpose, we use Malliavin calculus based on the Frechet differentiability in the directions of the reproducing kernel Hilbert space associated with fBm.
This paper considers a controlled Itô-Lévy process where the information available to the controller is possibly less than the overall information. All the system coefficients and the objective performance functional are allowed to be random, possibly non-Markovian. Malliavin calculus is employed to derive a maximum principle for the optimal control of such a system where the adjoint process is explicitly expressed.
Suppose $B$ is a Brownian motion and $B^n$ is an approximating sequence of rescaled random walks on the same probability space converging to $B$ pointwise in probability. We provide necessary and sufficient conditions for weak and strong $L^2$-convergence of a discretized Malliavin derivative, a discrete Skorokhod integral, and discrete analogues of the Clark-Ocone derivative to their continuous counterparts. Moreover, given a sequence $(X^n)$ of random variables which admit a chaos decomposition in terms of discrete multiple Wiener integrals with respect to $B^n$, we derive necessary and suff…
In this paper, we price the zero-coupon bond of the extended Cox-Ingersoll-Ross model by a Dyson type formula established in one of the authors' paper Jin, Peng and Schelllhorn (2016) using Malliavin calculus. This formula provides a fast convergent series to represent the bond price, and it depends on the given drift and volatility of the interest rate process but not the instantaneous forward rate used in Maghsoodi (1996). This expression can be also regarded as a new solution to a class of Reccati equations with time-dependent coefficients.
We investigate Stein-Malliavin approximations for nonlinear functionals of geometric interest of Gaussian random eigenfunctions on the unit $d$ -dimensional sphere ${\mathbb{S}}^{d},$ $d\geq 2.$ All our results are established in the high energy limit, i.e. for eigenfunctions corresponding to growing eigenvalues. More precisely, we provide an asymptotic analysis for the variance of random eigenfunctions, and also establish rates of convergence for various probability metrics for Hermite subordinated processes, arbitrary polynomials of finite order and square integral nonlinear transforms; the …
Malliavin calculus is implemented in the context of [M. Hairer, A theory of regularity structures, Invent. Math. 2014]. This involves some constructions of independent interest, notably an extension of the structure which accomodates a robust, and purely deterministic, translation operator, in $L^2$-directions, between "models". In the concrete context of the generalized parabolic Anderson model in 2D - one of the singular SPDEs discussed in the afore-mentioned article - we establish existence of a density at positive times.
Let G be a Lie group equipped with a set of left invariant vector fields. These vector fields generate a function ξon Wiener space into G via the stochastic version of Cartan's rolling map. It is shown here that, for any smooth function f with compact support, f(ξ) is Malliavin differentiable to all orders and these derivatives belong to L^p(μ) for all p>1, where μis Wiener measure.
In this article, we give some existence and smoothness results for the law of the solution to a stochastic heat equation driven by a finite dimensional fractional Brownian motion with Hurst parameter $H>1/2$. Our results rely on recent tools of Young integration for convolutional integrals combined with stochastic analysis methods for the study of laws of random variables defined on a Wiener space.
The discrete-time approximation for nonlinear filtering problems is related to both of strong and weak approximations of stochastic differential equations. In this paper, we propose a new method of proof for the convergence of approximate nonlinear filter analyzed by Jean Picard (1984), and show a more general result than the original one. For the proof, we develop an analysis of Hilbert space valued functionals on Wiener space.
The extremely useful method of Malliavin calculus has not yet gained adequate popularity because of the complicated analytic apparatus of this method. The author attempts here to propose a simplified algebraic formalism similar to Malliavin calculus, but based on the notion of creation-annihilation operators instead of Malliavin derivative to replace analytic theorems with algebraic computations. Three test problems: the valuation of portfolio with stochastic payoff function, the expression of the terminal payoff through stochastic integral and the approximate equation for the high-frequency m…
It is well known that Malliavin calculus can be applied to a stochastic differential equation with Lipschitz continuous coefficients in order to clarify the existence and the smoothness of the solution. In this paper, we apply Malliavin calculus to the CEV-type Heston model whose diffusion coefficient is non-Lipschitz continuous and prove the Malliavin differentiability of the model.
We study the asymptotic behavior as $n\to \infty$ of the sequence $$S_{n}=\sum_{i=0}^{n-1} K(n^α B^{H_{1}}_{i}) (B^{H_{2}}_{i+1}-B^{H_{2}}_{i})$$ where $B^{H_{1}}$ and $B^{H_{2}}$ are two independent fractional Brownian motions, $K$ is a kernel function and the bandwidth parameter $α$ satisfies certain hypotheses in terms of $H_{1}$ and $H_{2}$. Its limiting distribution is a mixed normal law involving the local time of the fractional Brownian motion $B^{H_{1}}$. We use the techniques of the Malliavin calculus with respect to the fractional Brownian motion.
