On Lévy processes conditioned to avoid zero

Pantí, Henry

doi:10.30757/alea.v14-32

Cited by 19 publications

(31 citation statements)

References 20 publications

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“…We consider the stable process killed on hitting 0. From Kyprianou et al [17] (for general self-similar Markov processes) or Pantí [21] and Yano [27] (for general Lévy processes) we know that |x| α−1 , x = 0, is harmonic for the killed transition semigroup…”

Section: Preliminariesmentioning

confidence: 99%

Stable processes conditioned to avoid an interval

Döring

Kyprianou

Weissmann

2020

Stochastic Processes and their Applications

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Conditioning Markov processes to avoid a domain is a classical problem that has been studied in many settings. Ingredients for standard arguments involve the leading order tail asymptotics of the distribution of the first hitting time of the domain of interest and its relation to an underlying harmonic function.In the present article we condition stable processes to avoid intervals. The required tail asymptotics in the stable setting for α ≥ 1 go back to classical work of Blumenthal et al. and Port from the 1960s. For α < 1, we appeal to recent results centred around the so-called deep factorisation of the stable process to compute hitting probabilities and, moreover, to identify the associated harmonic functions for all α ∈ (0, 2). With these in hand, we thus prove that conditioning to avoid an interval is possible in the classical sense and that the resulting process is a Doob h-transform of the stable process killed on entering the aforesaid interval. Appealing to the representation of the conditioned process as a Doob h-transform, we verify that the conditioned process is transient.

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Section: Preliminariesmentioning

confidence: 99%

Stable processes conditioned to avoid an interval

Döring

Kyprianou

Weissmann

2020

Stochastic Processes and their Applications

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“…Over the last decade, fluctuation theory stimulated the study of potential and spectral theory for symmetric Lévy processes (and in particular those with completely monotone jumps); see, e.g., [7,8,9,10,12,14,24,25,26,33,34,35,36,37,38,39,41,53,54,55,72] and the references therein. There are, however, very few papers where similar problems are studied for asymmetric processes, see [27,28,40,63,75]. The results of the present article may therefore stimulate the development of potential theory and spectral theory for asymmetric Lévy processes.…”

Section: Introductionmentioning

confidence: 71%

Fluctuation theory for Lévy processes with completely monotone jumps

Kwaśnicki¹

2019

Electron. J. Probab.

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We study the Wiener-Hopf factorization for Lévy processes X t with completely monotone jumps. Extending previous results of L.C.G. Rogers, we prove that the space-time Wiener-Hopf factors are complete Bernstein functions of both the spatial and the temporal variable. As a corollary, we prove complete monotonicity of: (a) the tail of the distribution function of the supremum of X t up to an independent exponential time; (b) the Laplace transform of the supremum of X t up to a fixed time T , as a function of T . The proof involves a detailed analysis of the holomorphic extension of the characteristic exponent f (ξ) of X t , including a peculiar structure of the curve along which f (ξ) takes real values.

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“…Note that this argument does not work if (−a, −1) is replaced by (1, a) because v 1 has a pole in 1. Now we plug in (15) and (16) to obtain for all ε > 0:…”

Section: 2mentioning

confidence: 99%