On Approximations of Small Jumps of Subordinators with Particular Emphasis on a Dickman-Type Limit

Abstract

Let X be a pure-jump subordinator (i.e. nondecreasing Lévy process with no drift) with infinite Lévy measure, let Xε be the sum of jumps not exceeding ε, and let µ(ε)=E[Xε(1)]. We study the question of weak convergence of Xε/µ(ε) as ε ↓0, in terms of the limit behavior of µ(ε)/ε. The most interesting case reduces to the weak convergence of Xε/ε to a subordinator whose marginals are generalized Dickman distributions; we give some necessary and sufficient conditions for this to hold. For a certain significant class of subordinators for which the latter convergence holds, and whose most prominent representative is the gamma process, we give some detailed analysis regarding the convergence quality (in particular, in the context of approximating X itself). This paper completes, in some respects, the study made by Asmussen and Rosiński (2001).

One-dimensional distributions of subordinators with upper truncated Lévy measure, and applications

“…A thorough characterization of the class of pure-jump subordinatorsX such that X ε /ε converges weakly to X c as ε → 0 is provided by the author in [4]. It turns out [4] that the Dickman limit plays a central role in the context of approximating small jumps of subordinators.…”

“…Moreover, since ε is assumed to be small and since the above series converges rapidly (depending, of course, on cT ), it is clear that we may truncate the series at some moderate number of terms. As for the CPPX ε , it has rate ∞ ε ρ(x) dx, which is asymptotically c log(1/ε) as ε → 0 (this is a particular case of a proposition in [4]), and is, hence, relatively small in general. This is very advantageous from a computational point of view.…”

The α-Dependence of Stochastic Differential Equations Driven by Variants of α-Stable Processes

“…More precisely, the Brownian motion W can be approximated by an appropriate renormalization of the compensated sum of small jumps of a given Lévy process, see Proposition 3.5 below. In the same spirit we mention the work [8] which completes, in some sense the previous one, where it is shown that the process {t, t ∈ [0, 1]} is a weak limit of a renormalized (in an appropriate sense) sum of small jumps of classes of subordinator. We note that the family {X…”

“…Condition (8) means that the corresponding Lévy process has finite variation paths. The characteristic exponent Ψ Λ T S α is given by…”

“…We denote the expectation of X ε (1) by µ(ε) := (0,ε] sdΛ(s). We consider the renormalized process Y ε := µ(ε) −1 X ε and state the following convergence result proved in [8].…”