The quadratic variation of continuous time stochastic processes in vector lattices

Abstract

We define and study order continuity, topological continuity, γ-Hölder-continuity and Kolmogorov–Čentsov-continuity of continuous-time stochastic processes in vector lattices and show that every such kind of continuous submartingale has a continuous compensator of the same kind. The notion of variation is introduced for continuous time stochastic processes and for a γ-Hölder-continuous martingale with finite variation, we prove that it is a constant martingale. The localization technique for not necessarily bounded martingales is introduced and used to prove our main result which states that the quadratic variation of a continuous-time γ-Hölder continuous martingale X is equal to its compensator 〈X〉.