Asymptotic properties of power variations of Lévy processes

By Jean Jacod. 2005

83 citations recorded by the source corpus.

We determine the asymptotic behavior of the realized power variations, and more generally of sums of a given function f evaluated at the increments of a Levy process between the successive times i Δn for i = 0,1,...,n . One can elucidate completely the first order behavior, that is the convergence in probability of such sums, possibly after normalization and/or centering: it turns out that there is a rather wide variety of possible behaviors, depending on the structure of jumps and on the chosen test function f . As for the associated central limit theorem, one can show some versions of it, but unfortunately in a limited number of cases only: in some other cases a CLT just does not exist.

Subfields: Finance, Management Science and Operations Research

View Paper: https://www.semanticscholar.org/paper/364474d8f88396dc469e1ca4f67a450fe374853b

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Citations in the corpus

Small-time moment asymptotics for Lévy processes (2008, background): “Jacod [13] proves the following more general result (see his Theorem 2.1 and equation (5.8)).”
Nonparametric Estimation for Lévy Models Based on Discrete-Sampling (2009, background): “Under the assumption of this Proposition, it turns out that limt→0 1t Γt(ϕ) = ν(ϕ 2) (see Lemma 5.5 in Jacod (2007)), and thus, σ̄2n,π := Var n∑ k=1 ξπk = 1 tn n∑ k=1 ΓΔk(ϕ) −→ ν(ϕ2), as the mesh π̄ := maxk {tk − tk−1} → 0.”
Detecting Jumps from Levy Jump Diffusion Processes (2010, unknown): “First notice that by a simple direct calculation – see also Jacod (2005), ∆t−1 K K∑ i=1 c2i I{|ci|≤2d} P−→ 0.”
Estimation for L\'{e}vy processes from high frequency data within a long time interval (2011, methodology): “The result is proved, for example, in Barndorff-Nielsen, Shephard and Winkel [(2006), Theorem 1, page 804] [see also Aı̈t-Sahalia and Jacod (2007)].”
Asymptotic properties of realized power variations and related functionals of semimartingales (2006, background): “em 2.12, after normalization and centering. When 2 <r≤ 3, there is no CLT, at least with the natural centering of the associated LLN, although a CLT with a more adequate centering might exist: see [8] for a more thorough description of this fact when Xis a L´evy process. ✷ Finally, we give the an”
Small-time expansions for the transition distributions of Lévy processes (2008, background): “This approach is based on the following recent asymptotic result by Jacod [10]: lim t→0 1 t E g(Xt) = σ2 + ∫ g(x)ν(dx), (1.9) valid for a ν-continuous bounded function g such that g(x) ∼ x2, as x → 0.”