The underlying Lévy process

1.1 The underlying Lévy process

We consider a R-valued Lévy process X = X t , t ≥ 0 starting from 0. We assume that X is the canonical process on the Skorohod space DR + , R of càd-làg real-valued paths, endowed with the canonical filtration. The law of the process X starting from 0 will be denoted by P and the corresponding expectation by E. Most of the following facts on Lévy processes can be found in [12]. In this paper, we assume that X • has no negative jumps, • has first moments, • is of infinite variation, • does not drift to +∞. The law of X is characterized by its Laplace transform: ∀λ ≥ 0, E ” e −λX t — = e t ψλ where, as X does not drift to + ∞, its Laplace exponent ψ can then be written as 1, where the Lévy measure π is a Radon measure on R + positive jumps that satisfies the integrability condition Z 0,+∞ ℓ ∧ ℓ 2 πdℓ +∞ X has first moments, the drift coefficient α is non negative X does not drift to +∞ and β ≥ 0. As we ask for X to be of infinite variation, we must additionally suppose that β 0 or R 0,1 ℓ πdℓ = +∞. As X is of infinite variation, we have, see Corollary VII.5 in [12], lim λ→∞ λ ψλ = 0. 9 Let I = I t , t ≥ 0 be the infimum process of X , I t = inf ≤s≤t X s , and let S = S t , t ≥ 0 be the supremum process, S t = sup ≤s≤t X s . We will also consider for every 0 ≤ s ≤ t the infimum of X over [s, t]: I s t = inf s ≤r≤t X r . We denote by J the set of jumping times of X : J = {t ≥ 0, X t X t − } 10 and for t ≥ 0 we set ∆ t := X t − X t − the height of the jump of X at time t. Of course, ∆ t 0 ⇐⇒ t ∈ J . The point 0 is regular for the Markov process X − I, and −I is the local time of X − I at 0 see [12], chap. VII. Let N be the associated excursion measure of the process X − I away from 0, and let 1436 σ = inf{t 0; X t − I t = 0} be the length of the excursion of X − I under N. We will assume that under N, X = I = 0. Since X is of infinite variation, 0 is also regular for the Markov process S − X . The local time L = L t , t ≥ 0 of S − X at 0 will be normalized so that E[e −λS L−1 t ] = e −tψλλ , where L −1 t = inf{s ≥ 0; L s ≥ t} see also [12] Theorem VII.4 ii.

1.2 The height process