= Exponential martingale of a Lévy process
{title2=$M_t^u=\exp(iuX_t-t\psi(u))$}
With the convention $\mathbb E e^{iuX_t}=e^{t\psi(u)}$, this complex process is a <martingale> for each real $u$. Its absolute value is the deterministic finite number $e^{-t\operatorname{Re}\psi(u)}$. Conditioning on the past at time $s$ and using <independent increments> multiplies $e^{iuX_s-t\psi(u)}$ by $e^{(t-s)\psi(u)}$, giving $M_s^u$. Under the opposite exponent convention the sign of the compensating term reverses.
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