Itô's rule and Lévy's theorem in vector lattices

Abstract

The change of variable formula, or Itô's rule, is studied in a Dedekind complete vector lattice E with weak order unit E. Using the functional calculus we prove that for a Hölder continuous semimartingale Xt=Xa+Mt+Bt,t∈J, and a twice continuously differentiable function f, the formula(0.1)f(Xt)=f(Xa)+∫0tf′(Xs)dMs+∫0tf′(Xs)dBs+12∫0tf″(Xs)d〈M〉s,0≤s≤t∈J holds. The first integral in the formula is an Itô integral with reference to the local martingale M and the second and third integrals are Dobrakov-type integrals of a vector valued function with reference to a vector valued measure. Using the formula, we prove Lévy's characterization of Brownian motion as being a continuous martingale with compensator tE. The proof of this result yields a concrete description of abstract Brownian motion defined in vector lattices.