Two-parameter Stochastic Processes With Finite Variation

Abstract:

Let E be a Banach space with norm ||, and f: R2+ ?E a function with finite variation. Properties of the variation are studied, and an associated increasing real-valued function |f| is defined.

Sufficient conditions are given for f to have properties analogous to those of functions of one variable. A correspondence f ??f between such functions and E-valued Borel measures on R2+ is established, and the equality | ?f |= ?|f| is proved. Correspondences between E-valued two-parameter processes X with finite variation |x| and E-valued stochastic measures with finite variation are established. The case where X takes values in L(E,F) (F a Banach space) is studied, and it is shown that the associated measure ?x takes values in L(E,F"); some x sufficient conditions for y to be L(E,F)-valued are given. Similar results for the converse problem are established, and some conditions sufficient for the equality | ?x |= ?|x| are given.

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