Simplified calculus for semimartingales: Multiplicative compensators and changes of measure

Aleš Černý and Johannes Ruf

Stochastic Processes and their Applications, 2023, vol. 161, issue C, 572-602

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 when such compensation is meaningful. This generalization of the Lévy–Khintchin formula fills an existing gap in the literature. It allows, for example, the computation of the Mellin transform of a signed stochastic exponential, which in turn has practical applications in mean–variance portfolio theory. Girsanov-type results based on multiplicatively compensated semimartingales simplify treatment of absolutely continuous measure changes. As an example, we obtain the characteristic function of log returns for a popular class of minimax measures in a Lévy setting.

Keywords: Girsanov; Lévy–Khintchin; Mellin transform; Predictable compensator; Process with independent increments; Semimartingale representation (search for similar items in EconPapers)
Date: 2023
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Working Paper: Simplified calculus for semimartingales: Multiplicative compensators and changes of measure (2023)
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DOI: 10.1016/j.spa.2023.04.010

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