Stability of Sigma-Martingale Densities in L log L Under an Equivalent Change of Measure

Abstract

An equivalent sigma-martingale measure (E-sigma-MM) for a given stochastic process S is a probability measure R equivalent to the original measure P such that S is an R-sigma-martingale. Existence of an E-sigma-MM is equivalent to a classical absence-of-arbitrage property of S, and is invariant if we replace the reference measure P with an equivalent measure Q. Now suppose that there exists an E-sigma-MM for S such that the density dR/dP is in L log L(P). Does this property also remain invariant if we replace P by some equivalent Q? We prove that the answer is Yes if one imposes instead of a global only a local integrability requirement.