The hitting time of zero for a stable process

Kuznetsov, Alexey; Kyprianou, Andreas E.; Pardo, Juan Carlos; Watson, Alexander R.

doi:10.1214/ejp.v19-2647

Cited by 43 publications

(76 citation statements)

References 34 publications

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“…The next theorem generalises its counterpart for positive self-similar Markov processes, namely Theorem 2.2 and is due to Chaumont et al (2013) and Kuznetsov et al (2014). …”

Section: Proposition 32 (♥)mentioning

confidence: 77%

“…where the expectation on the right-hand side is known to be infinite; see for example Kuznetsov et al (2014). That said, if we put together some of the ingredients we have examined in the preceding computations in the right way, we can deduce the following precise result which does not seem to be known in the existing literature.…”

Section: 2mentioning

confidence: 96%

“…This is stipulated by the strong law of large numbers for MAPs (see again Chapter XI.2 of Asmussen, 2003) and the behaviour of the leading eigenvalue χ as a function of γ, for which the proposition below is lifted from Proposition 3.4 of Kuznetsov et al (2014). Note that, since Ψ(0) = Q, it is always the case that χ(0) = 0 and v(0) = (1, · · · , 1).…”

Section: Proposition 34 (♥)mentioning

confidence: 99%

“…With only the sporadic works of e.g. Graversen and Vuolle-Apiala (1986); Vuolle-Apiala and ; Kiu (1980) in the 1980s, only very recently has the bigger picture begun to emerge some 30 or more years later in Xiao (1998); Nane et al (2010); Chaumont et al (2013); Kuznetsov et al (2014); Alili et al (2017); Kyprianou et al (2015); Kyprianou (2016); ).…”

Section: Onward Researchmentioning

confidence: 99%

“…This point of view offers deeper probabilistic insights into some of the aforementioned potential analytical relations; see for example Bertoin and Yor (2002); Bertoin and Caballero (2002); Caballero and Chaumont (2006a,b); Chaumont et al (2009);Patie (2009Patie ( , 2012; Bogdan and Żak (2006); Patie (2012); ; Kuznetsov et al (2014); ; Kuznetsov and Pardo (2013); Kyprianou (2016); ; Alili et al (2017).…”

mentioning

confidence: 98%

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Stable Lévy processes, self-similarity and the unit ball

Kyprianou¹

2018

ALEA

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“…The next theorem generalises its counterpart for positive self-similar Markov processes, namely Theorem 2.2 and is due to Chaumont et al (2013) and Kuznetsov et al (2014). …”

Section: Proposition 32 (♥)mentioning

confidence: 77%

Section: 2mentioning

confidence: 96%

Section: Proposition 34 (♥)mentioning

confidence: 99%

Section: Onward Researchmentioning

confidence: 99%

mentioning

confidence: 98%

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Stable Lévy processes, self-similarity and the unit ball

Kyprianou¹

2018

ALEA

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Unimodality of Hitting Times for Stable Processes

Letemplier

Simon

2014

Lecture Notes in Mathematics

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Abstract. We show that the hitting times for points of real α−stable Lévy processes (1 < α ≤ 2) are unimodal random variables. The argument relies on strong unimodality and several recent multiplicative identities in law. In the symmetric case we use a factorization of Yano et al. [15], whereas in the completely asymmetric case we apply an identity of the second author [11]. The method extends to the general case thanks to a fractional moment evaluation due to Kuznetsov et al. [6]. Introduction and statement of the resultA real random variable X is said to be unimodal if there exists a ∈ R such that its distribution function P[X ≤ x] is convex on (−∞, a) and concave on (a, +∞). When X is absolutely continuous, this means that its density is non-decreasing on (−∞, a] and nonincreasing on [a, +∞). The number a is called a mode of X and might not be unique. A random variable with a single mode is called strictly unimodal. The problem of unimodality has been intensively studied for infinitely divisible random variables and the reader can consult to Chapter 10 in [10] for details and references. This problem has also been settled in the framework of hitting times of processes and Rösler -see Theorem 1.2 in [9] -showed that hitting times for points of real-valued diffusions are always unimodal. However, much less is known when the underlying process has jumps, for example when it is a Lévy process.In this paper we consider a real strictly α−stable process (1 < α ≤ 2), which is a Lévy process {X t , t ≥ 0} starting from zero and having characteristic exponentwhere ρ ∈ [1 − 1/α, 1/α] is the positivity parameter of {X t , t ≥ 0} that is ρ = P[X 1 ≥ 0]. We refer to [16] and to Chapter 3 in [10] for an account on stable laws and processes. In particular, comparing the parametrisations (B) and (C) in the introduction of [16] shows that the characteristic exponent of {X t , t ≥ 0} takes the more familiar formwith ρ = 1/2 + (1/πα) tan −1 (θ tan(πα/2)) and c = cos(πα(ρ − 1/2)). The constant c is a scaling parameter which could take any arbitrary positive value without changing our purposes below. We are interested in the hitting times for points of {X t , t ≥ 0}: τ x = inf{t > 0, X t = x}, x ∈ R.2000 Mathematics Subject Classification. 60E05, 60G18, 60G51, 60G52.

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Kac–Lévy Processes

Ratanov

2018

J Theor Probab

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Markov-modulated Lévy processes with two different regimes of restarting are studied. These regimes correspond to the completely renewed process and to the process of Markov modulation, accompanied by jumps. We give explicit expressions for the Lévy-Khintchine exponent in the case of a two-state underlying Markov chain.For the renewal case, the limit distributions (as t → ∞) are obtained. In the case of processes with jumps, we present some results for the exponential functional.

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The hitting time of zero for a stable process

Cited by 43 publications

References 34 publications

Stable Lévy processes, self-similarity and the unit ball

Stable Lévy processes, self-similarity and the unit ball

Unimodality of Hitting Times for Stable Processes

Kac–Lévy Processes

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