  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 L\'evy--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 L\'evy setting.
