Multiple Ito Integrals Research Articles

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

Gaussian and their Subordinated Self-similar Random Generalized Fields

A large class of generalized random fields is defined, containing random elements $F$ of $\mathscr{J}'$, where $\mathscr{J}'$ is the dual of the Schwartz space $\mathscr{J} = \mathscr{J}(\mathbb{R}^\nu)$. Such a generalized random field is translation-invariant if $F\phi$ is the same as $F\psi$ for any translate $\psi$ of $\phi$; it is invariant under the renormalization group with index $_\kappa$ (or self-similar with index $_\kappa$) if $F\phi_\lambda = \lambda^{-\alpha}F\phi$ for all $\lambda > 0$ and $\phi \in \mathscr{L}$, where $\phi_\lambda$ is the rescaled test function $\phi_\lambda(x) = \lambda^{-\nu}\phi(x/\lambda)$. Recent work of several authors has shown that self-similar generalized random fields on $\mathbb{R}^\nu$, and self-similar random fields on $\mathbb{Z}^\nu$ which can be constructed from them, arise naturally in problems of statistical mechanics and limit laws of probability theory. They generalize the theory of stable distributions. Here the class of all translation-invariant self-similar Gaussian generalized random fields on $\mathbb{R}^\nu$ is completely described. By the discretization of such fields the class of self-similar Gaussian fields with discrete arguments (found by Sinai) is extended. Finally, a class of generalized random fields subordinated to the self-similar translation-invariant Gaussian ones is constructed. These non-Gaussian generalized random fields are Wick powers (multiple Ito integrals) of the Gaussian ones.

Convergence of integrated processes of arbitrary Hermite rank

Let {X(s), −∞<s<∞} be a normalized stationary Gaussian process with a long-range correlation. The weak limit in C[0,1] of the integrated process $$Z_x \left( t \right) = \frac{1}{{d\left( x \right)}}\mathop \smallint \limits_0^{xt} G\left( {X\left( s \right)} \right)ds,{\text{ }}x \to \infty$$ , is investigated. Here d(x) = x H L(x) with $$\frac{1}{2}$$ <H<1 and L(x) is a slowly varying function at infinity. The function G satisfies EG(X(s))=0, EG 2 (X(s))<∞ and has arbitrary Hermite rank m≧1. (The Hermite rank of G is the index of the first non-zero coefficient in the expansion of G in Hermite polynomials.) It is shown thatZ x (t) converges for all m≧1 to some process ¯Z m (t) that depends essentially on m. The limiting process ¯Z m (t) is characterized through various representations involving multiple Ito integrals. These representations are all equivalent in the finite-dimensional distributions sense. The processes ¯Z m (t) are non-Gaussian when m≧2. They are self-similar, that is,¯Z m (at) and a H ¯Z m (t) have the same finite-dimensional distributions for all a>0.