Simplified Calculus for Semimartingales: Multiplicative Compensators and Changes of Measure

Abstract

The paper develops multiplicative compensation for complex-valued semimartingales and studies some of its consequences. It is shown that the stochastic exponential of any complex-valued semimartingale with independent increments becomes a true martingale after multiplicative compensation, where such compensation is meaningful. This generalization of the Lévy-Khintchin formula fills an existing gap in the literature. We further report Girsanov-type results based on non-negative multiplicatively compensated semimartingales. In particular, we obtain a simplified expression for the multiplicative compensator under the new measure.

Keywords: Girsanov, Lévy-Khintchin, Mellin transform, predictable compensator, process with independent increments, semimartingale representation

JEL Classification: C65, G11