Quadratic variation for cylindrical martingale-valued measures

Abstract

This article focuses in the definition of a quadratic variation for cylindrical orthogonal martingale-valued measures defined on Banach spaces. Sufficient and necessary conditions for the existence of such a quadratic variation are provided. Moreover, several properties of the quadratic variation are explored, as the existence of a quadratic variation operator. Our results are illustrated with numerous examples and in the case of a separable Hilbert space, we delve into the relationship between our definition of quadratic variation and the intensity measures defined by Walsh (1986) for orthogonal martingale measures with values in separable Hilbert spaces. We finalize with a construction of a quadratic covariation and we explore some of its properties.