Integration of random functions with respect to a random Wiener measure. (Russian) Zbl 0577.60056

Baklan, V. V.

Teor. Veroyatn. Mat. Stat. 29, 10-14 (1983).

Let W be a random Wiener measure, given on a measurable space (T,\({\mathcal M},\lambda (dt))\). Let \(\{\phi_ k\}\) be a basis of \(L_ 2(T,\lambda)\), \(r\) the \(\sigma\)-algebra generated by the random variables \(x_ k=\int_{T}\phi_ k(t)W(dt)\), and f a \(r\)-measurable function. Here the stochastic integral \(I(f)=\int_{T}f(t,x)W(dt)\) is defined as \(I(f):=\sum^{\infty}_{k=1}f(t,x)\phi_ k(t)x_ k\). Conditions for the convergence of this series are given.

Reviewer: C.Baldauf

MSC:

60H05	Stochastic integrals
60J65	Brownian motion

Keywords:

Wiener measure; stochastic integral