Simplified Calculus for Semimartingales: Multiplicative Compensators and Changes of Measure

arXiv preprint 2006.12765

26 Pages Posted: 23 Jun 2020 Last revised: 12 Apr 2021

See all articles by Aleš Černý

Aleš Černý

Bayes Business School (formerly Cass), City, University of London

Johannes Ruf

London School of Economics & Political Science (LSE) - London School of Economics

Date Written: June 22, 2020

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

Suggested Citation

Černý, Aleš and Ruf, Johannes, Simplified Calculus for Semimartingales: Multiplicative Compensators and Changes of Measure (June 22, 2020). arXiv preprint 2006.12765, Available at SSRN: https://ssrn.com/abstract=3633622 or http://dx.doi.org/10.2139/ssrn.3633622

Aleš Černý (Contact Author)

Bayes Business School (formerly Cass), City, University of London

Northampton Square
London, EC1V 0HB
United Kingdom

Johannes Ruf

London School of Economics & Political Science (LSE) - London School of Economics ( email )

United Kingdom

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