Stochastic integrals with respect to Gaussian random measures. (Russian) Zbl 0749.60051

Dorogovtsev, A. A.

Teor. Veroyatn. Mat. Stat., Kiev 44, 56-63 (1991).

The article is devoted to the construction of the integral \(\int_ X f(u)\xi(du)\) in the case when \(\xi\) is a Gaussian random measure with independent values on disjoint sets and \(f\) is a random function. There exist two different well-known constructions of such an integral: extended and symmetric stochastic integrals. These integrals have different domains of definition: subsets of \(L_ 2(\Omega\times X,P\times\lambda)\), where \(\lambda(du)=M\xi^ 2(du)\), and may have different values on those random functions, for which they are well- defined. So it is interesting to find such class of random functions for which the definition of the integral is natural and to study its relations with two main definitions. In the article the family of “simple” random functions is considered. The functions of this family have the following property: their values on some subsets of \(X\) do not depend on values of \(\xi\) on these subsets. For “simple” functions stochastic integral can be defined as a special case of symmetric integral but has some properties of Itô integral. It is proved that the extended stochastic integral (after localization without finite second moment) can be received as a limit of integrals from “simple” functions.

Reviewer: A.A.Dorogovtsev (Kiev)

MSC:

60H05

Stochastic integrals

Keywords:

Gaussian random measure; extended and symmetric stochastic integrals