Stochastic integrals with respect to generalized random orthogonal measures in Banach spaces. (English. Russian original) Zbl 0625.60008

Ponomarenko, A. I.

Theory Probab. Math. Stat. 33, 103-110 (1986); translation from Teor. Veroyatn. Mat. Stat. 33, 92-99 (1985).

The author constructs the integral \(\int_{S}\Phi (ds)B(s)\), where \(\Phi\) is a measure on a measurable space (S,\({\mathcal B})\) with values in the space L(X,H) of continuous linear operators from a Banach space X into a Hilbert space H (which is sometimes realized as the space of scalar random variables of second order) and such that \(\Phi '(\Delta_ 1)\Phi (\Delta_ 2)=\mu (\Delta_ 1\cap \Delta_ 2)V\), where \(\mu\) is a finite positive measure on \({\mathcal B}\) and V is a linear symmetric positive operator from X into the dual space X’; B(\(\cdot)\) is a strongly measurable function on S with values in L(Y,X) (Y is another Banach space) such that \(\int_{S}\| B(s)\|^ 2d\mu (s)<\infty.\)
The author uses this construction to define the stochastic integral with respect to a generalized Wiener measure (defined on an n-dimensional parallelepiped with values in L(X,H)). Further an analogue of stochastic differential equations is investigated and some conditions of existence of solutions are given.

Reviewer: S.A.Chobanjan

MSC:

60B11	Probability theory on linear topological spaces
60H05	Stochastic integrals
60G57	Random measures
28C20	Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.)
60G12	General second-order stochastic processes

Keywords:

orthogonally scattered measure; linear symmetric positive operator; strongly measurable function; generalized Wiener measure; stochastic differential equations