Multiparameter processes with stationary increments: Spectral representation and integration

Abstract

In this article, a class of multiparameter processes with wide-sense stationary increments is studied. The content is as follows. (1) The spectral representation is derived; in particular, necessary and sufficient conditions for a measure to be a spectral measure is given. The relations to a commonly used class of processes, studied e.g. by Yaglom, is discussed. (2) Some classes of deterministic integrands, here referred to as predomains, are studied in detail. These predomains consist of functions or, more generally, distributions. Necessary and sufficient conditions for completeness of the predomains are given. (3) In a framework covering the classical Walsh-Dalang theory of a temporal-spatial process which is white in time and colored in space, a class of predictable integrands is considered. Necessary and sufficient conditions for completeness of the class are given, and this property is linked to a certain martingale representation property.