(X_{t})_{t \geq 0}

with

X_{0} = 0

almost surely, independent increments, stationary increments, and stochastic continuity. The latter means

X_{t} - X_{s} \to 0

in probability as

t \to s

. Its usual path convention is càdlàg: paths are right-continuous and have left limits. Equivalently one starts with the increment and continuity-in-probability conditions and then chooses the càdlàg version.

The Lévy characteristic exponent is specified here by

E e^{i θ X_{t}} = e^{- t ψ (θ)} .

(1)

Some conventions use

e^{t ψ (θ)}

instead, with

ψ = - ψ

. The sign convention must be fixed before computing the exponent. The zero process satisfies this definition; excluding it requires an additional nondegeneracy condition.