Brief tutorial of Lévy processes
نویسنده
چکیده
Some fundamental properties related to Lévy processes are discussed. Topics include infinitely divisible distributions, Lévy-Khintchine formula, Poisson random measures, Lévy-Itô decomposition, series representations, and density transformations. 1 Basic properties • In short, a Lévy process X = {Xt}t≥1 is a R-valued process with independent and stationary increments whose paths are right-continuous with left-limits and that has no fixed jump-times. • The law of X is uniquely determined by the distribution of X1 (Why? ). • Clearly, the marginal distributions are infinitely-divisible: – ξ is said to be infinitely divisible if for any n, one can construct n iid r.v. ξ1, . . . , ξn such that ξ D = ξ1 + · · ·+ ξn. – It has a close relation to limits in distribution of a array of row-wise i.i.d. r.v.: ∗ ξ is infinitely divisible iff there exists {ξn,k}n k=1, i.i.d. for each n, such that kn ∑ k=1 ξn,k D −→ ξ, as n →∞. – It has an explicit characteristic function: ∗ ξ is infinitely divisible iff Ee = exp { − 2 〈u,Au〉+ i 〈b, u〉+ ∫ Rd ( ei〈u,x〉 − 1− i 〈u, x〉1|x|≤1 ) ν(dx) } ,
منابع مشابه
Lévy processes
We give a brief introduction to the class of stochastic processes known as Lévy processes, concentrating principally on their relation with infinitely divisible distributions and the Lévy-Itô decomposition.
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