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arXiv · 2006.12765

Simplified calculus for semimartingales: Multiplicative compensators and changes of measure

Aleš Černý·Johannes Ruf
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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.

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Aleš Černý, Johannes Ruf. 2023-05-09. Simplified calculus for semimartingales: Multiplicative compensators and changes of measure. https://doi.org/10.1016/j.spa.2023.04.010

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