## Chaos decomposition

### The Malliavin derivative based on chaos expansion: Definition, motivation and properties

Bernt Øksendal (Universitetet i Oslo)

For Brownian motion there is a simple and natural introduction to the Malliavin derivative as a stochastic gradient in the Hida white noise probability space. It can be proved that this definition is actually equivalent to a definition based on the Wiener-Itô chaos expansion theorem. Each of these two interpretations has its advantages, and by exploiting the equivalence between them one can easily deduce the basic properties of this operator.

For (discontinuous) Lévy processes the situation is different. The two approaches are no longer equivalent, and one has to make a choice regarding how to define the Malliavin derivative for such processes. We will argue that the chaos expansion approach is better, because it leads to a Malliavin derivative operator which is more useful in applications. For example, it gives an easy proof of a natural L ́evy process extension of the Clark-Ocone formula, which is important in mathematical finance.

### Malliavin calculus without probability and application to Lévy processes

Dave Applebaum (University of Sheffield)

The key concepts of Malliavin calculus associated to a process having a chaotic decomposition are the gradient and divergence operators acting in duality. In this talk we start with an abstract Fock space and develop the properties of the universal annihilation and creation processes. Using a natural isomorphism with the $L^{2}$ space of a general Lévy process we transform the universal annihilation and creation operators into the gradient and divergence (respectively) and obtain the Itô-Skorohod isometry in that space.

### Stein’s method and Malliavin calculus on the Poisson space

Giovanni Peccati (Université du Luxembourg)

We show how to combine Stein’s method and Malliavin calculus on the Poisson space, in order to obtain explicit bounds in (possibly multidimensional) central limit theorems. Based on joint works with J.L. Solé (Barcelona), M.S. Taqqu (Boston), F. Utzet (Barcelona) and C. Zheng (Paris).

### On Sobolev spaces of pure jump Lévy Functionals

John Hosking (I.N.R.I.A.)

In essence this talk is a deliberation on the relationship, and differences, between two approaches to a Malliavin calculus for pure jump Lévy functionals (PJLFs); that of the chaos expansion approach, and that of J. Picard’s mass transformation based approach. This is a matter that has been studied by A. Løkka in the $L^{2}$-setting; where too the work of D. Applebaum on a universal Mallavin calculus is also relevant. We address the question of whether the Malliavin calculus type operators from the Picard approach may give rise to a family of bounded linear operators on a PJLF Sobolev space framework which is defined in an analogous manner to Watanabe’s system of Wiener functional Sobolev spaces. We report that such an extension is achievable in the restriction to the $L^{2}$-setting. However, we also demonstrate that this full program is not feasible in the $L^{p}$ case for certain $p \in (1, \infty) \backslash \{ 2 \}$. Indeed, we present a counter-example to a corresponding form of Kre-Meyer type inequality in this PJLF setting.

### Malliavin’s type derivative for Lévy processes: the chaos expansion point of view

Josep Vives (Universitat de Barcelona)

K. Itô showed in [2] that the space of square integrable functionals of a centered independent random measure that extends a L ́evy process enjoys the so called chaotic representation property, generalizing his previous similar result for the Brownian motion [1]. In other words, this means that this space of square integrable functionals has Fock space structure. This fact allows to develop a formal chaotic calculus using algebraic gradient (anihilation) and diver- gence (creation) operators. In [3] and [4] a probabilistic interpretation of the gradient operator for the standard Poisson case was given. And its extension to the general Lévy case was done in [5], [6] and [7]. This two last papers will be the main body of my talk. Moreover I will talk about some connections with other constructions of a Mallliavin type calculus for Lévy processes based on different points of view and some applications to Finance (pricing and hedging, computation of sensitivities, etc.).

#### References:

[1] K. Itô (1951): Multiple Wiener Integral. Journal of Mathematical Society of Japan 3: 157- 169.
[2] K. Itô (1956): Spectral type of shift transformation of differential processes with stationary increments. Transactions of the American Mathematical Society 81: 252-263.
[3] D. Nualart, J. Vives (1990): Anticipative calculus for the Poisson process based on the Fock space. Séminaire de Probabilités 24: 154-165.
[4] D. Nualart, J. Vives (1995): A dualitiy formula on the Poisson space and some applications. Proceedings on the Ascona Conference on Stochastic Analysis. Progress in Probability. Birkhauser.
[5] F. E. Benth, G. Di Nunno, A. Løkka, B. Øksendal and F. Proske (2003): Explicit represen-tations of the minimal variance portfolio in market driven by Lévy processes. Mathematical Finance 13: 55-72.
[6] J. L. Solé, F. Utzet and J. Vives (2007): Chaos expansions and Malliavin calculus for Lévy processes. Stochastic Analysis and Applications: The Abel Symposium 2007 (2): 595-612. Springer. [7] J. L. Solé, F. Utzet and J. Vives (2007): Canonical Lévy process and Malliavin calculus. Stochastic Processes and their Applications 117: 165- 187,2.

### Wavelet variations of non-linear subordinated processes with memory

Ciprian Tudor (Université Lille 1)

Consider a stationary Gaussian sequence $\{X_{t}\}_{t\in\mathbb{Z}}$ with long-range dependence as input of a non-linear filter, with output $G(X_{t})$ assumed to be mean-squared integrable. We focus on the wavelet coefficients  $\{W_{j,k}, j\ge 0, k\in \mathbb{Z}\}$ of the output and the corresponding scalogram $S_{n,j}= \sum_{k=0}^{n-1} W_{j,k} ^{2}$. Our goal is to study the asymptotic behavior of the process $\{S_{n,j+m}, m\in \mathbb{Z}\}$ as both the sample size $n$ and the scale $j$ goes to infinity. The limit, it turns out, is either fractional Brownian motion, the non-Gaussian Rosenblatt process or a linear combination thereof. And unexpectedly, the nature of the limit depends on the sequence of non-vanishing Hermite coefficients of the deterministic function $G$ and not merely on the order of the first non-vanishing Hermite coefficient. The proofs are based on chaos expansion into multiple stochastic integrals. This is a joint work with M. Clausel (Lyon), F. Roueff (Telecom Paris) and M. Taqqu (Boston).