We establish two results about local times of spectrally positive stable processes. The first is a general approximation result, uniform in space and on compact time intervals, in a model where each jump of the stable process may be marked by a random path. The second gives moment control on the Hölder constant of the local times, uniformly across a compact spatial interval and in certain random time intervals. For the latter, we introduce the notion of a Lévy process restricted to a compact interval, which is a variation of Lambert's Lévy process confined in a finite interval and of Pistorius' doubly reflected process. We use the results of this paper to exhibit a class of path-continuous branching processes of Crump-Mode-Jagers type with continuum genealogical structure. A further motivation for this study lies in the construction of diffusion processes in spaces of interval partitions and R-trees, which we explore in forthcoming articles.
We study the distribution of the area under the normalized excursion of a spectrally positive stable L{é}vy process L, as well as the area under its meander, and under L conditioned to stay positive. Our results involve a special case of Wright's function, which may be seen as a generalization of the classic Airy function appearing in similar Brownian's areas.
For a spectrally positive and strictly stable process with index in (1,2), a series representation is obtained for the joint distribution of the "first passage triple" that consists of the time of first passage and the undershoot and the overshoot at first passage. The result leads to several corollaries, including 1) the joint law of the first passage triple and the pre-passage running supremum, and 2) at a fixed time point, the joint law of the process' value, running supremum, and the time of the running supremum. The representation can be decomposed as a sum of strictly positive functions …
For a spectrally positive strictly stable process with index in (1,2), the paper obtains i) the density of the time when the process makes first exit from an interval by hitting the interval's lower end point before jumping over its upper end point, and ii) the joint distribution of the time, the undershoot, and the jump of the process when it makes first exit the other way around. For i), the density of the time of first exit is expressed as an infinite sum of functions, each the product of a polynomial and an exponential function, with all coefficients determined by the roots of a Mittag-Lef…
De Finetti's optimal dividend problem has recently been extended to the case dividend payments can only be made at Poisson arrival times. This paper considers the version with bail-outs where the surplus must be nonnegative uniformly in time. For a general spectrally negative Lévy model, we show the optimality of a Parisian-classical reflection strategy that pays the excess above a given barrier at each Poisson arrival times and also reflects from below at zero in the classical sense.
We investigate the upper tail probabilities of the all-time maximum of a stable Lévy process with a power negative drift. The asymptotic behaviour is shown to be exponential in the spectrally negative case and polynomial otherwise, with explicit exponents and constants. Analogous results are obtained, at a less precise level, for the fractionally integrated stable Lévy process. We also study the lower tail probabilities of the integrated stable Lévy process in the presence of a power positive drift.
We propose a positivity preserving implicit Euler-Maruyama scheme for a jump-extended Cox-Ingersoll-Ross (CIR) process where the jumps are governed by a compensated spectrally positive $α$-stable process for $α\in (1,2)$. Different to the existing positivity preserving numerical schemes for jump-extended CIR or CEV (Constant Elasticity Variance) process, the model considered here has infinite activity jumps. We calculate, in this specific model, the strong rate of convergence and give some numerical illustrations. Jump extended models of this type were initially studied in the context of branc…
We establish a uniform Hausdorff dimension result for the inverse image sets of real-valued strictly $α$-stable Lévy processes with $1< α\le 2$. This extends a theorem of Kaufman for Brownian motion. Our method is different from that of Kaufman and depends on covering principles for Markov processes.
We prove a metric space scaling limit for a critical random graph with independent and identically distributed degrees having power-law tail behaviour with exponent $α+1$, where $α\in (1,2)$. The limiting components are constructed from random $\mathbb{R}$-trees encoded by the excursions above its running infimum of a process whose law is locally absolutely continuous with respect to that of a spectrally positive $α$-stable Lévy process. These spanning $\mathbb{R}$-trees are measure-changed $α$-stable trees. In each such $\mathbb{R}$-tree, we make a random number of vertex-identifications, who…
We consider a stable Cox--Ingersoll--Ross process driven by a standard Wiener process and a spectrally positive strictly stable Lévy process, and we study asymptotic properties of the maximum likelihood estimator (MLE) for its growth rate based on continuous time observations. We distinguish three cases: subcritical, critical and supercritical. In all cases we prove strong consistency of the MLE in question, in the subcritical case asymptotic normality, and in the supercritical case asymptotic mixed normality are shown as well. In the critical case the description of the asymptotic behavior of…
