Multiple Stochastic Integrals Research Articles

Stochastic integrals: a combinatorial approach

A combinatorial definition of multiple stochastic integrals is given in the setting of random measures. It is shown that some properties of such stochastic integrals, formerly known to hold in special cases, are instances of combinatorial identities on the lattice of partitions of a set. The notion of stochastic sequences of binomial type is introduced as a generalization of special polynomial sequences occuring in stochastic integration, such as Hermite, Poisson–Charlier and Kravchuk polynomials. It is shown that identities for such polynomial sets have a common origin.

Discretization of stochastic differential equations by the product expansion for the chen series

Following the approach of Sussmann [16] to expand the Chen series as a product of an exponential Lie series based on a Philip Hall basis, this paper constructs the Stratonovich-Taylor-Hall (STH) discretization schemes of both integral and fractional orders for stochastic differential equations. The STH schemes are numerically efficient in the sense that they involve only the minimum numbers of multiple stochastic integrals. General multiple stochastic integrals are regarded as systems of stochastic differential equations that can be solved by lower order schemes. Convergence of the STH schemes is proved in the almost sure sense by means of the discrete stochastic Gronwall inequality

Symmetrization and concentration inequalities for multilinear forms with applications to zero-one laws for Lévy chaos

We consider stochastic processes $X=\{X(t),t\in T\}$ represented as a Lévy chaos of finite order, that is, as a finite sum of multiple stochastic integrals with respect to a symmetric infinitely divisible random measure. For a measurable subspace V of $R\sp T$ we prove a very general zero-one law $P(X\in V)=0$ or 1, providing a complete analogue to the corresponding situation in the case of symmetric infinitely divisible processes (single integrals with respect to an infinitely divisible random measure). Our argument requires developing a new symmetrization technique for multi-linear Rademacher forms, as well as generalizing Kanter's concentration inequality to multiple sums.

Multiple stochastic integrals appearing in the stochastic Taylor expansions

Multiple stochastic integrals appearing in the stochastic Taylor expansions

Limit theorems for some polynomial statistics of the poisson process

The central limit theorem and the theorem on large deviations for the functionals of the Poisson random process are proved. The formulas for cumulants of multiple stochastic integrals (m.s.i.) with respect to the Poisson process are obtained. The m.s.i. may be considered as anU-statistics arising in queueing theory as well as a generalization of the well-known Poisson shot-noise process, having wide applications.

Chaotic and variational calculus in discrete and continuous time for the poisson process

We study a new interpretation of the Poisson space as a triplet (H,B,P) where if is a Hilbert spaceB is the completion of H and P is the extension to the Borel σ-algebra of B of a cylindrical measure on B. A discrete chaotic decomposition of L 2(B P) is defined, along with multiple stochastic integrals of elements of H on. It turns out that the directional derivative of functionals in L 2(B P) in the direction of an element of H is an annihilation operator on the discrete chaotic decomposition. By composition with the Poisson process, we deduce continuous-time operators of derivation and divergence that form the number operator on the discrete chaotic decomposition. Those results are applied to the representation of random variables in the Wiener-Poisson chaotic decomposition

Asymptotic Distributions for Weighted $U$-Statistics

We prove limit theorems for weighted $U$-statistics and express the limit by means of multiple stochastic integrals. This is a generalization of the paper by O'Neil and Redner. In that paper the method of moments was applied which does not work in the general case. Hence we had to work out a different method. In particular, in Theorem 4 we describe the limit of a model proposed by O'Neil and Redner. In this model the weight functions cause an intricate cancellation, and the limit can be presented as a sum of multiple stochastic integrals with different multiplicities.

Spherical level-crossing measures for chi-square random fields

Spherical measures of exceeding a low level are considered for strongly dependent chi-square fields. The limit distributions are sums of multiple stochastic integrals.

Generalized functionals of Brownian motion

In this paper we discuss some recent developments in the theory of generalized functionals of Brownian motion. First we give a brief summary of the Wiener‐Ito multiple Integrals. We discuss some of their basic properties, and related functional analysis on Wiener measure space. then we discuss the generalized functionals constructed by Hida. The generalized functionals of Hida are based on L2‐Sobolev spaces, thereby, admitting only Hs, s ∈ R valued kernels in the multiple stochastic integrals. These functionals are much more general than the classical Wiener‐Ito class. The more recent development, due to the author, introduces a much more broad class of generalized functionals which are based on Lp‐Sobolev spaces admitting kernels from the spaces 𝒲p,s, s ∈ R. This allows analysis of a very broad class of nonlinear functionals of Brownian motion, which can not be handled by either the Wiener‐Ito class or the Hida class. For s ≤ 0, they represent generalized functionals on the Wiener measure space like Schwarz distributions on finite dimensional spaces. In this paper we also introduce some further generalizations, and construct a locally convex topological vector space of generalized functionals. We also present some discussion on the applications of these results.

Series expansions of multiple L�vy integrals

Multiple stochastic integrals with respect to an infinitely divisible symmetric random measure without a Gaussian component admit LePage-type representations by means of certain multiple random series.

The approximation of multiple stochastic integrals

A method for approximating the multiple stochastic integrals appearing in stochaslic Taylor expansions is proposed. It is based on a series expansion of the Brownian bridge process. Some higher order time discrete approximations for the simulation of Ito processes using these approximate multiple stochastic integrals arc also included.

Multiple Stochastic Integrals with Respect to Symmetric Infinitely Divisible Random Measures

Multiple stochastic integrals with respect to an infinitely divisible symmetric random measure are constructed for integrands taking values in a Banach space.

On theL2-Theory of product stochastic measures and multiple wiener–ito integrals

In this work we construct product stochastic measures using symmetric tensor products in appropriate Hilbert spaces and then construct multiple Wiener-Itô integrals using the vector valued measure approach. A clear relationship between the theories of L 2 -multiple stochastic integrals and vector valued measures is established. This approach yields, as special cases, the multiple stochastic integrals of Itô Surgailis, Ogura, and Fox and Taqqu, among others

Relations between multiple ito and stratonovich integrals

In this paper we investigate relations within and between sets of multiple Ito and Stratonovich integrals which are useful in the application of stochastic Taylor expansions of Ito processes. We obtain formulae expressing multiple Ito integrals in terms of multiple Stratonovich integrals and vice versa and also formulae relating multiple stochastic integrals of the same type

Time Series Regression With a Unit Root and Infinite-Variance Errors

In [4] Chan and Tran give the limit theory for the least-squares coefficient in a random walk with i.i.d. (identically and independently distributed) errors that are in the domain of attraction of a stable law. This paper discusses their results and provides generalizations to the case of I(1) processes with weakly dependent errors whose distributions are in the domain of attraction of a stable law. General unit root tests are also studied. It is shown that the semiparametric corrections suggested by the author in other work [22] for the finite-variance case continue to work when the errors have infinite variance. Surprisingly, no modifications to the formulas given in [22] are required. The limit laws are expressed in terms of ratios of quadratic functional of a stable process rather than Brownian motion. The correction terms that eliminate nuisance parameter dependencies are random in the limit and involve multiple stochastic integrals that may be written in terms of the quadratic variation of the limiting stable process. Some extensions of these results to models with drifts and time trends are also indicated.

Multiple integration with respect to Poisson and Lévy processes

Necessary and sufficient conditions are given for the existence of a multiple stochastic integral of the form ∫...∫fdX 1...dXd, where X 1, ..., X d are components of a positive or symmetric pure jump type Levy process in ℝd. Conditions are also given for a sequence of integrals of this type to converge in probability to zero or infinity, or to be tight. All arguments proceed via reduction to the special case of Poisson integrals.

A survey of numerical methods for stochastic differential equations

The development of numerical methods for stochastic differential equations has intensified over the past decade. The earliest methods were usually heuristic adaptations of deterministic methods, but were found to have limited accuracy regardless of the order of the original scheme. A stochastic counterpart of the Taylor formula now provides a framework for the systematic investigation of numerical methods for stochastic differential equations. It suggests numerical schemes, which involve multiple stochastic integrals, of higher order of convergence. We shall survey the literature on these and on the earlier schemes in this paper. Our discussion will focus on diffusion processes, but we shall also indicate the extensions needed to handle processes with jump components. In particular, we shall classify the schemes according to strong or weak convergence criteria, depending on whether the approximation of the sample paths or of the probability distribution is of main interest.

Spherical measures of sojourn above a high level for one class of random fields

We study spherical measures of sojourn above a high level for strongly dependent chi-square type random fields (when the integral of the field correlation function is divergent). Limiting distributions are specified by multiple stochastic integrals.

About A. N. Kolmogorov's work on the entropy of dynamical systems

In the fall of 1957 A. N. Kolmogorov started lecturing on the theory of dynamical systems and supervised a seminar on the same theme at the Mechanical–Mathematical Department of Moscow State University. He began his lectures with the theory of systems with a pure point spectrum, which he approached from a probabilistic point of view. This approach, undoubtedly, has many advantages. In the seminar we studied Ito's theory of multiple stochastic integrals and, under the supervision of A. N. Kolmogorov, I. V. Girsanov constructed an example of a Gaussian dynamical system with a simple continuous spectrum. At one of the meetings of the seminar, still before the advent of entropy, Kolmogorov suggested a proof of an assertion, which today would read: the unitary operator induced by a K-automorphism has a countable Lebesgue spectrum. At this point Kolmogorov was studying Shannon's theory of information and the concept of the capacity of functional spaces. Judging by his well known article, we can say that the first of these, and all that is related to it, played a big role in the development of information theory in our country. The investigation of capacity is connected with Kolmogorov's work on Hilbert's 13th problem and was summarized in his well-known survey co-authored by V. M. Tihomirov.

On the tail behaviour of the distribution function of multiple stochastic integrals

Let F n (u) denote the empirical distribution function of a sample of i.i.d. random variables with uniform distribution on [0, 1]. Define $$\bar \mu _n^* (u) = \sqrt n [F_n (u) - u]$$ , and consider the integrals $$I(t) = \int\limits_0^t {\int\limits_0^1 {...\int\limits_0^1 {f(u_1 ,...,u_s )} } }$$ where f is a bounded measurable function. We give a good upper bound on the probability $$P{\text{ }}(\mathop {\sup }\limits_{0 \leqq t \leqq 1} {\text{ | }}I{\text{(}}t{\text{) |}} \geqq x)$$ . An analogous estimate is given for multiple integrals with respect to a Poisson process.