The set-indexed Ito integral

Abstract

We construct an Ito-type stochastic integral where the integrator is a process indexed by a semilattice of compact subsets of a fixed topological spaceT and the integrands, which are indexed by the points inT, possess a natural form of predictability. The definition of the integral involves, among other things, an Ito-type isometry defined in terms of the set-indexed quadratic variation of the integrator. The martingale property and quadratic variation for the resulting integral process are derived. In addition, employing the notion of stopping set from Ivanoff and Merzbach (1995), we construct and study a set-indexed local integral. A novel and flexible notion of predictability for set-indexed processes is defined and characterized, permitting the integration of a set-indexed integrand against a set-indexed process.