Constructing multiple stochastic integrals on non-Gaussian product measures

We study a construction of multiple stochastic integrals of nonrandom functions with respect to the product measures generated by stochastic processes admitting representations as multiple orthogonal random series. This construction is compared with some classical schemes of constructing stochastic integrals of such a kind.     

Generalized multiple stochastic integrals and the representation of wiener functionals

In this paper we study a generalized multiple stochastic integral for non-adapted integrands following Skorohod's approach. The main properties of this integral are derived. In particular, we prove a Fubini type result and discuss the relation of this multiple integral to the Malliavin calculus. It turns out that this integral includes other kinds of multiple stochastic integrals like those of Hajek and Wong. Finally, we apply these results to the representation of functionals of the multiparameter Wiener process, obtaining explicit formulas for the kernels of the representation in terms of conditional expectations of Malliavin derivatives     

On stochastic integration by series of Wiener integrals

Stochastic integrals of random functions with respect to a white-noise random measure are defined in terms of random series of usual Wiener integrals. Conditions for the existence of such integrals are obtained in terms of the nuclearity of certain operators onL ² -spaces. The relation with the Fisk-Stratonovich symmetric integral is also discussed.This research was supported by AFOSR Contract No. F49620 82 C 0009.     

On multiple integrals of special form

Multiple integrals generalizing the iterated kernels of integral operators are expressed as single integrals in the case of a special representation of the kernel (this is our theorem). Besides integral equations, Markov processes involve these integrals as well. As a consequence of the theorem, we obtain transition probability densities of certain Markov processes. As an illustration, we consider nine examples.     

Central limit theorems for multiple stochastic integrals and Malliavin calculus

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 also give a new proof of the main theorem in [D. Nualart, G. Peccati, Central limit theorems for sequences of multiple stochastic integrals, Ann. Probab. 33 (2005) 177–193] using techniques of Malliavin calculus. Finally, we extend our result to the multidimensional case and prove a weak convergence result for a sequence of square integrable random vectors, giving an application.     

On the Gaussian approximation of vector-valued multiple integrals

By combining the findings of two recent, seminal papers by Nualart, Peccati and Tudor, we get that the convergence in law of any sequence of vector-valued multiple integrals F_n towards a centered Gaussian random vector N, with given covariance matrix C, is reduced to just the convergence of: (i) the fourth cumulant of each component of F_n to zero; (ii) the covariance matrix of F_n to C. The aim of this paper is to understand more deeply this somewhat surprising phenomenon. To reach this goal, we offer two results of a different nature. The first one is an explicit bound for d(F,N) in terms of the fourth cumulants of the components of F, when F is a R^d-valued random vector whose components are multiple integrals of possibly different orders, N is the Gaussian counterpart of F (that is, a Gaussian centered vector sharing the same covariance with F) and d stands for the Wasserstein distance. The second one is a new expression for the cumulants of F as above, from which it is easy to derive yet another proof of the previously quoted result by Nualart, Peccati and Tudor.     

On the limit behaviour of the joint distribution function of order statistics

Fork ₀ fixed we consider the joint distribution functionF _n ^k of then-k smallest order statistics ofn real-valued independent, identically distributed random variables with arbitrary cumulative distribution functionF. The main result of the paper is a complete characterization of the limit behaviour ofF _n ^k (x ₁,,x _n-k) in terms of the limit behaviour ofn(1-F(x _n)) ifn tends to infinity, i.e., in terms of the limit superior, the limit inferior, and the limit if the latter exists. This characterization can be reformulated equivalently in terms of the limit behaviour of the cumulative distribution function of the (k+1)-th largest order statistic. All these results do not require any further knowledge about the underlying distribution functionF.     

On the equivalence of Beukers-type and Sorokin-type multiple integrals

It is well known that a triple Beukers-type integral, as defined by G. Rhin and C. Viola, can be transformed into a suitable triple Sorokin-type integral. I will discuss possible extensions to the n-dimensional case of a similar equivalence between suitably defined Beukers-type and Sorokin-type multiple integrals, with consequences on the arithmetical structure of such integrals as linear combinations of zeta-values with rational coefficients.