Entrance and exit at infinity for stable jump diffusions

Abstract

In his seminal work from the 1950s, William Feller classified all one-dimensional diffusions on $-\infty \leq a<b\leq \infty $ in terms of their ability to access the boundary (Feller’s test for explosions) and to enter the interior from the boundary. Feller’s technique is restricted to diffusion processes as the corresponding differential generators allow explicit computations and the use of Hille–Yosida theory. In the present article, we study exit and entrance from infinity for the most natural generalization, that is, jump diffusions of the form \begin{equation*}dZ_{t}=\sigma (Z_{t-})\,dX_{t},\end{equation*} driven by stable Lévy processes for $\alpha \in (0,2)$. Many results have been proved for jump diffusions, employing a variety of techniques developed after Feller’s work but exit and entrance from infinite boundaries has long remained open. We show that the presence of jumps implies features not seen in the diffusive setting without drift. Finite time explosion is possible for $\alpha \in (0,1)$, whereas entrance from different kinds of infinity is possible for $\alpha \in [1,2)$. Accordingly, we derive necessary and sufficient conditions on $\sigma $.

Our proofs are based on very recent developments for path transformations of stable processes via the Lamperti–Kiu representation and new Wiener–Hopf factorisations for Lévy processes that lie therein. The arguments draw together original and intricate applications of results using the Riesz–Bogdan–Żak transformation, entrance laws for self-similar Markov processes, perpetual integrals of Lévy processes and fluctuation theory, which have not been used before in the SDE setting, thereby allowing us to employ classical theory such as Hunt–Nagasawa duality and Getoor’s characterisation of transience and recurrence.