Moment estimates for Lévy Processes

For real L´evy processes ( X t ) t ≥ 0 having no Brownian component with Blumenthal-Getoor index β , the estimate E sup s ≤ t | X s − a p s | p ≤ C p t for every t ∈ [0 , 1] and suitable a p ∈ R has been established by Millar [6] for β < p ≤ 2 provided X 1 ∈ L p . We derive extensions of these estimates to the cases p > 2 and p ≤ β .


Introduction and results
We investigate the L p -norm (or quasi-norm) of the maximum process of real Lévy processes having no Brownian component.A (càdlàg) Lévy process X = (X t ) t≥0 is characterized by its so-called local characteristics in the Lévy-Khintchine formula.They depend on the way the "big" jumps are truncated.We will adopt in the following the convention that the truncation occurs at size 1.So that E e iuXt = e −tΨ(u) with Ψ(u) = −iua + 1 2 σ2u2 − (e iux − 1 − iux1 {|x|≤1} )dν(x) (1.1) where u, a ∈ R, σ2 ≥ 0 and ν is a measure on R such that ν({0}) = 0 and x2 ∧ 1dν(x) < +∞.
The measure ν is called the Lévy measure of X and the quantities (a, σ2, ν) are referred to as the characteristics of X.One shows that for p > 0, E |X (see [7]).The index β of the process X introduced in [2] is defined by {|x|≤1} |x| p dν(x) < +∞}.(1.3)In the sequel we will assume that σ2 = 0, i.e. that X has no Brownian component.Then the Lévy-It decomposition of X reads x(µ − λ ⊗ ν)(ds, dx) + t 0 {|x|>1} xµ(ds, dx) (1.4) where λ denotes the Lebesgue measure and µ is the Poisson random measure on R + × R associated with the jumps of X by [4] , [7]).
Theorem 1 Let (X t ) t≥0 be a Lévy process with characteristics (a, 0, ν) and index β such that If X 1 is symmetric one observes that Y = X since the symmetry of X 1 implies a = 0 and the symmetry of ν (see [7]).We emphasize that in view of the Kolmogorov criterion for continuous modifications the above bounds are best possible as concerns powers of t.In case p > β and p ≤ 2, these estimates are due to Millar [6].However, the Laplace-transform approach in [6] does not work for p > 2. Our proof is based on the Burkholder-Davis-Gundy inequality.
For the case p < β we need some assumptions on X. Recall that a measurable function This means that ϕ(1/x) is regularly varying at infinity with index −b.Slow variation corresponds to b = 0.
Important special cases are as follows.
Corollary 1.1 Assume the situation of Theorem 2 (with ν symmetric if β = 1) and let U denote any of the processes X, Y, (X t − at) t≥0 .
Theorem 3 Let (X t ) t≥0 be a Lévy process with characteristics (a, 0, ν) and index β such that (1.6) where the process Y is defined as in Theorem 2.
The above estimates are optimal (see Section 3).The paper is organized as follows.Section 2 is devoted to the proofs of Theorems 1, 2 and 3. Section 3 contains a collection of examples.

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. As concerns X (1) , consider the martingale where µ 1 denotes the Poisson random measure associated with the jumps of X (1) .The starting idea is to part the 'small' and the 'big' jumps of X (1) in a non homogeneous way with respect to the function s → s 1/β .Indeed one may decompose Z (1) as follows and are martingales.Observe that for every q > 0 and t ≥ 0, Consequently, where ψ(t) := t 0 x1 {|x|>s 1/β } dν 1 (x)ds.Furthermore, for every r > β or r = 2 and t ≥ 0 In the sequel let C denote a finite constant that may vary from line to line.We first claim that for every t ≥ 0, r ∈ (β, 2] ∩ [1, 2] and for r = 2, In fact, it follows from the Burkholder-Davis-Gundy inequality and from p/r ≤ 1, r/2 ≤ 1 that Exactly as for M , one gets for every t ≥ 0 and every q ∈ If ν is symmetric then (2.4) holds for every q ∈ [p, 2] (which of course provides additional information in case p < 1 only).Indeed, ψ = 0 by the symmetry of ν so that and for q ∈ [p, 1] (2.5) In the case β < 1 we consider the process Exactly as in (2.5) one shows that for t ≥ 0 and r ∈ (β, 1] (2.9) If ν is symmetric then Y = Z = (X t − at) t≥0 and (2.9) is valid for every r ∈ (β, 2], q ∈ [p, 2].Now we deduce Theorem 2. Assume p ∈ (0, β) and (1.5).The constant c in the above decomposition of X is specified by the constant from (1.5).Then one just needs to investigate the integrals appearing in the right hand side of the inequalities (2.7) -(2.10).One observes that Theorem 1.5.11 in [1] yields for r > β, which in turn implies that for small t, Similarly, for 0 < q < β, Using (2.2) for the case β = 2 and t + t p = o(t p/β l(t) α ) as t → 0, α > 0, for the case β > 1 one derives Theorem 2.
1 | p < +∞ if and only if E |X t | p < +∞ for every t ≥ 0 and this in turn is equivalent to E sup s≤t |X s | p < +∞ for every t ≥ 0. Furthermore, E |X 1 | p < +∞ if and only if 6) is satisfied with β = 1.Using (1.2) one observes that E |X 1 | p < +∞ for every p > 0. It follows from Theorems 1 and 3 that s≤t |X s | p = O(t) if p > 1, E sup s≤t |X s | p = O((t (− log t)) p ) if p ≤ 1.If γ = 0, then ν is symmetric and hence Theorem 2 yieldsE sup s≤t |X s | p = O(t p ) if p < 1.