Recent zbMATH articles in MSC 60E10https://zbmath.org/atom/cc/60E102024-03-13T18:33:02.981707ZWerkzeugJoint occupation times in an infinite interval for spectrally negative Lévy processes on the last exit timehttps://zbmath.org/1528.600222024-03-13T18:33:02.981707Z"Li, Yingqiu"https://zbmath.org/authors/?q=ai:li.yingqiu"Wei, Yushao"https://zbmath.org/authors/?q=ai:wei.yushao"Hu, Yangli"https://zbmath.org/authors/?q=ai:hu.yangliLévy processes are a kind of processes with stationary independent increments and play a very important role in many fields. A spectrally negative Lévy process (SNLP) is a Lévy process without positive jumps. With the Markov property and infinite divisibility, it plays an important role in the study of financial risks. The occupation time of an SNLP plays an important role in the study of bankruptcy problems. The authors of this article use the Poisson approach and perturbation approach to find some joint Laplace transforms of the last exit time by their joint occupation times over semiinfinite intervals \((-\infty,0)\) and \((0,\infty)\) by virtue of the associated scale functions.
Reviewer: Ze-Chun Hu (Chengdu)Rudin extension theorems on product spaces, turning bands, and random fields on balls cross timehttps://zbmath.org/1528.600522024-03-13T18:33:02.981707Z"Porcu, Emilio"https://zbmath.org/authors/?q=ai:porcu.emilio"Feng, Samuel F."https://zbmath.org/authors/?q=ai:feng.samuel-f"Emery, Xavier"https://zbmath.org/authors/?q=ai:emery.xavier"Peron, Ana P."https://zbmath.org/authors/?q=ai:peron.ana-paulaSummary: Characteristic functions that are radially symmetric have a dual interpretation, as they can be used as the isotropic correlation functions of spatial random fields. Extensions of isotropic correlation functions from balls into \(d\)-dimensional Euclidean spaces, \(\mathbb{R}^d\), have been understood after Rudin. Yet, extension theorems on product spaces are elusive, and a counterexample provided by Rudin on rectangles suggests that the problem is challenging. This paper provides extension theorems for multiradial characteristic functions that are defined in balls embedded in \(\mathbb{R}^d\) cross, either \(\mathbb{R}^{d'}\) or the unit sphere \(\mathbb{S}^{d'}\) embedded in \(\mathbb{R}^{d'+1}\), for any two positive integers \(d\) and \(d'\). We then examine Turning Bands operators that provide bijections between the class of multiradial correlation functions in given product spaces, and multiradial correlations in product spaces having different dimensions. The combination of extension theorems with Turning Bands provides a connection with random fields that are defined in balls cross linear or circular time.