Locally Ф-integrable σ-martingale densities for general semimartingales

Abstract

A P-sigma-martingale density for a given stochastic process S is a local P-martingale Z>0 starting at 1 such that the product ZS is a P-sigma-martingale. Existence of a P-sigma-martingale density is equivalent to a classic absence-of-arbitrage property of S, and it is invariant if we replace the reference measure P with a locally equivalent measure Q. Now suppose that there exists a P-sigma-martingale density for S. Can we find another P-sigma-martingale density for S having some extra local integrability I_loc(P) under P? We show that the answer is always positive for one part of S that we identify, and we show that the complete answer depends in a precise quantitative way on the local integrability of the drift-to-jump ratio of the remaining "jumpy" part of S. Our proofs provide in addition new ideas and results in infinite-dimensional spaces.