Stability of the overshoot for Lévy processes
Let \(X_t\) be a real-valued Lévy process (i.e., a stochastically continuous random process with stationary and independent increments) with characteristic exponent \(\psi\) defined by \(\mathbb{E}e^{i\xi X_t} = e^{-t\psi(\xi)}\). Further, let \((\gamma,\sigma^2,\Pi)\) be the Lévy triplet which uniquely characterises \(\psi\) via the Lévy-Khinchin formula. The authors study the behaviour of the overshoot of the process upon exiting a strip resp. a halfspace. Denote by \[ T(r) = \inf\{t>0: |X_t|> r\} \quad\text{resp.}\quad T^*(r) = \inf\{t>0: X_t > r\} \] the first exit time of \(X_t\) from a symmetric strip resp.\ a halfplane. The authors etablish necessary and sufficient criteria for the following (almost sure or in probability) limits \[ \lim_{r\downarrow 0} \frac{|X(T(r))|}{r} = 1 \quad\text{and}\quad \lim_{r\downarrow 0} \frac{X(T^*(r))}{r} = 1 \tag{\(*\)} \] to happen. Similar limits for the two-sided exit problem are valid also when \(r\to\infty\), the proofs for this case are only sketched and resemble in scope and method the results for random walks given in the paper by \textit{P. S. Griffin} and \textit{R. A. Maller} [Adv. Appl. Probab. 30, 181-196 (1998; Zbl 0905.60064)]. It is shown that the first limit in \((*)\) holds in probability if and only if either (a) \(X_t\) lies in the domain of attraction of the normal law, that is, if \(X_t/b(t) \to N(0,1)\) in law as \(t\downarrow 0\) for some deterministic \(b(t)\) or if (b) \(X_t\) is relatively stable, i.e., if \(X_t/b(t) \to \pm 1\) in probability as \(t\downarrow 0\) for some deterministic \(b(t)\). (This explains the word stability in the title of the paper.) The criteria for both limits in \((*)\) to take place almost surely are given in terms of (functions of) the Lévy triple, using in particular certain equivalents of truncated mean and variance of the process and (one-sided) tails of the Lévy measure \(\Pi\). The proofs use two main ingredients: formulae for the distribution that the process crosses level \(\pm r\) by a jump upon exiting (they can be proved using point-process techniques and the associated Lévy system) and by now classical estimates for the supremum of a Lévy process and the first passage time which are due to \textit{W. E. Pruitt} [Ann. Probab. 9, 948-956 (1981; Zbl 0477.60033)].
scientific article; zbMATH DE number 1906049