Characteristic exponent of a Lévy process ($\Psi (u)$)

Use the conventionThen $\Psi$ is the characteristic exponent of the Lévy process. Some authors instead write $e^{t\psi (u)}$, with $\psi =-\Psi$. The sign convention should always be specified. The Lévy–Khintchine formula describes all possible exponents.

With the convention $\mathbb{E}{e}^{iu{X}_t}=e^{t\psi (u)}$, this complex process is a martingale for each real $u$. Its absolute value is the deterministic finite number $e^{-t\mathrm{Re}\psi (u)}$. Conditioning on the past at time $s$ and using independent increments multiplies $e^{iu{X}_s-t\psi (u)}$ by $e^{(t-s)\psi (u)}$, giving $M_s^u$. Under the opposite exponent convention the sign of the compensating term reverses.

Independent increments and stationary increments make a Lévy process's characteristic functions multiplicative in time. Together with continuity of Lévy characteristic functions and the value one at time zero, this forces an exponential. A continuous local complex logarithm is additive: its failure of additivity would be a continuous integer multiple of $2\pi i$, hence zero. The Cauchy functional equation and subdivision then give the exponential at every time. This construction of the characteristic exponent of a Lévy process precedes the Lévy–Khintchine formula describing the exponent's possible form.

For a Lévy process, stochastic continuity gives continuity of its characteristic function in time at every fixed frequency. The elementary estimate $|{\phi }_{X_s}(u)-{\phi }_{X_t}(u)|\le |u|\delta +2\mathbb{P}(|X_s-X_t|>\delta )$ proves this by first taking $s\to t$ and then $\delta \downarrow 0$. The argument works for any stochastically continuous stochastic process and needs no finite-moment assumption.