Operator bimeasures and stochastic integrals in normed spaces. (English. Russian original) Zbl 0840.60051

Ponomarenko, O. I.

Theory Probab. Math. Stat. 47, 129-137 (1993); translation from Teor. Jmovirn. Mat. Stat. 47, 129-139 (1992).

The author considers random measures and positive definite bimeasures on measure spaces \((\Lambda, {\mathcal A})\), resp., \((\Lambda \times \lambda, {\mathcal A} \times {\mathcal A}) \) with values in spaces of linear, resp., antilinear bounded mappings of \(X'\) into a Hilbert space \(H\), resp., into \(X''\) \((X\) is a normed linear space). An operator semivariation for (bi-)measures is introduced and properties thereof are studied. Stochastic integrals for (bi-)measures are defined, first for operator-valued step functions, then for bounded and unbounded operator-valued functions. The stochastic integral enjoys the usual properties. As an application, an integral representation theorem for \(L(X', H)\)-valued random functions is stated (without proof). No examples are given.

Reviewer: René L.Schilling (Erlangen)

MSC:

60H05	Stochastic integrals
60B99	Probability theory on algebraic and topological structures
46G10	Vector-valued measures and integration
28B05	Vector-valued set functions, measures and integrals

Keywords:

operator-valued bimeasure; operator-valued stochastic integral