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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