Stochasticities. Lévy processes. Part 2.

Define the characteristic function of X_t:

Φ_t(z) ≡ Φ_Xt(z) ≡ E[e^iz.X_t], z ∈ R^d

For t > s, by writing X_t+s = X_s + (X_t+s − X_s) and using the fact that X_t+s − X_s is independent of X_s, we obtain that t ↦ Φ_t(z) is a multiplicative function.

Φ_t+s(z) = Φ_Xt+s(z) = Φ_Xs(z) Φ_Xt+s − X_s(z) = Φ_Xs(z) Φ_Xt(z) = Φ_sΦ_t

The stochastic continuity of t ↦ X_t implies in particular that X_t → X_s in distribution when s → t. Therefore, Φ_Xs(z) → Φ_Xt(z) when s → t so t ↦ Φ_t(z) is a continuous function of t. Together with the multiplicative property Φ_s+t(z) = Φ_s(z).Φ_t(z), this implies that t ↦ Φ_t(z) is an exponential function.

Let (X_t)_t≥0 be a Lévy process on R^d. ∃ a continuous function ψ : R^d ↦ R called the characteristic exponent of X, such that:

E[e^iz.X_t] = e^tψ(z), z ∈ R^d

ψ is the cumulant generating function of X₁ : ψ = Ψ_X1 and that the cumulant generating function of X_t varies linearly in t: Ψ_Xt = tΨ_X1 = tψ. The law of X_t 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 X_t for a single time (say, t = 1).

This lecture covers stochastic processes, including continuous-time stochastic processes and standard Brownian motion by Choongbum Lee

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