White noise approach to stochastic integration
The purpose of this paper is to extend the well-known Ito integral \(\int^{b}_{a}\Phi (t,\omega)dB(t,\omega)\) with respect to a Brownian motion to not necessarily nonanticipating stochastic processes \(\Phi\) by using the white noise calculus. Starting from the formal equality \[ \int^{b}_{a}\Phi (t,\omega)dB(t,\omega)=\int^{b}_{a}\dot B(t,\omega)\Phi (t,\omega)dt \] the authors define two kinds of extended stochastic integrals, where they use two kinds of white noise multiplications introduced in the paper. In the nonanticipating case one of these integrals coincides with the usual Ito integral. After proving the existence and some basic properties of the introduced integrals the authors solve the following problem concerning the series expansion of the Ito integral: Let \((e_ k)_{k=1,2,...}\) be an orthonormal basis in \(L_ 2(a,b)\). If \(\int^{b}_{a}| \Phi (t,\omega)|\) 2 dt\(<\infty\omega\)-almost surely, then \(\Phi\) can be expanded by \(\Phi (t,\omega)=\sum^{\infty}_{k=1}a_ k(\omega)e_ k(t)\). Under which conditions does the series \(\sum^{\infty}_{k=1}a_ k(\omega)\int^{b}_{a}e_ k(t)dB(t,\omega)\) converge and, if it does, is the limit just the Ito integral? The authors give conditions on \(\Phi\) generating the convergence of the series and show that the limit is an extended Stratonovich integral defined in the paper. Note that the authors do not compare their integral with other extensions of the Ito integral known from the literature.
scientific article; zbMATH DE number 4036835