Define the characteristic function of Xt:
Φt(z) ≡ ΦXt(z) ≡ E[eiz.Xt], z ∈ Rd
For t > s, by writing Xt+s = Xs + (Xt+s − Xs) and using the fact that Xt+s − Xs is independent of Xs, we obtain that t ↦ Φt(z) is a multiplicative function.
Φt+s(z) = ΦXt+s(z) = ΦXs(z) ΦXt+s − Xs(z) = ΦXs(z) ΦXt(z) = ΦsΦt
The stochastic continuity of t ↦ Xt implies in particular that Xt → Xs in distribution when s → t. Therefore, ΦXs(z) → ΦXt(z) when s → t so t ↦ Φt(z) is a continuous function of t. Together with the multiplicative property Φs+t(z) = Φs(z).Φt(z), this implies that t ↦ Φt(z) is an exponential function.
Let (Xt)t≥0 be a Lévy process on Rd. ∃ a continuous function ψ : Rd ↦ R called the characteristic exponent of X, such that:
E[eiz.Xt] = etψ(z), z ∈ Rd
ψ is the cumulant generating function of X1 : ψ = ΨX1 and that the cumulant generating function of Xt varies linearly in t: ΨXt = tΨX1 = tψ. The law of Xt is therefore determined by the knowledge of the law of X1 : the only degree of freedom we have in specifying a Lévy process is to specify the distribution of Xt for a single time (say, t = 1).
This lecture covers stochastic processes, including continuous-time stochastic processes and standard Brownian motion by Choongbum Lee