1 Lévy Processes and Infinite Divisibility - Department of ...

ExercisesExercise 1 Using Definition 1.1, show that the sum oftwo (orindeed any finitenumber of) independent Lévy processes is again a Lévy process.Exercise 2 Suppose that S = {S n : n ≥ 0} is any random walk 9 and Γ p is anindependent random variable with a geometric distribution on {0,1,2,...} withparameter p.(i) Show that Γ p is infinitely divisible.(ii) Show that S Γp is infinitely divisible.Exercise 3 [ProofofLemma2.1]InthisexercisewederivetheFrullaniidentity.(i) Show for any function f such that f ′ exists and is continuous and f(0) andf(∞) are finite, that∫ ∞(f(ax)−f(bx)bdx = (f(0)−f(∞))log ,xa)0where b > a > 0.(ii) By choosing f(x) = e −x , a = α > 0 and b = α−z where z < 0, show that1(1−z/α) β = e−∫ ∞0 (1−ezx ) β x e−αx dxand hence by analytic extension show that the above identity is still validfor all z ∈ C such that Rz ≤ 0.Exercise 4 Establishing formulae (2.5) and (2.6) from the Lévy measure givenin (2.7) is the result of a series of technical manipulations of special integrals.In this exercise we work through them. In the following text we will use thegamma function Γ(z), defined byΓ(z) =∫ ∞0t z−1 e −t dtfor z > 0. Note the gamma function can also be analytically extended so thatit is also defined on R\{0,−1,−2,...} (see [16]). Whilst the specific definitionof the gamma function for negative numbers will not play an important role inthis exercise, the following two facts that can be derived from it will. For z ∈R\{0,−1,−2,...} the gamma function observes the recursion Γ(1+z) = zΓ(z)and Γ(1/2) = √ π.9 Recall that S = {S n : n ≥ 0} is a random walk if S 0 = 0 and for n = 1,2,3,... theincrements S n −S n−1 are independent and identically distributed.27