Vector-valued stochastic processes. V. Optional and predictable variation of stochastic measures and stochastic processes

Abstract

Let μ \mu be a stochastic measure, with values in a Banach space E E , with finite variation | μ | |\mu | . If μ \mu is optional (resp. predictable), then | μ | |\mu | is also optional (resp. predictable) provided E E is separable, or the dual of a separable space, or has the Radon-Nikodym property. Let A A be a right continuous stochastic process with values in E E , with finite variation | A | |A| . If A A is measurable (resp. optional, predictable), then | A | |A| , the continuous part | A | c |A{|^c} and the discrete part | A | d |A{|^d} have the same property.