The transformation theorem for two-parameter pure jump martingales

Summary LetM be a martingale of pure jump type, i.e. the compensation of the process describing the total of the point jumps ofM in the plane.M can be uniformly approximated by martingales of bounded variation jumping only on finitely many axial parallel lines. Using this fact we prove a change of variables formula in which forC ⁴-functions f the processf(M) is described by integrals off ^(k) (M),k=1, 2, with respect to stochastic integrators of the types expected: a martingale, two processes behaving as martingales in one direction and as processes of bounded variation in the other, and one process of bounded variation. Hereby we are led to investigate two types of random measures not considered so far in this context. By combination with the integrators already known, they might complete the set needed for a general transformation formula.