On the Laws of First Hitting Times of Points for One-Dimensional Symmetric Stable Lévy Processes

Yano, Kouji; Yano, Yuko; Yor, Marc

doi:10.1007/978-3-642-01763-6_8

Cited by 28 publications

(31 citation statements)

References 38 publications

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“…The symmetric case. Formula (5.12) and Lemma 2.17 in [16] yield together with the normalization (4.1) therein, which is the same as ours, the following independent factorization…”

Section: Proof Of the Theoremsupporting

confidence: 60%

“…For the sake of clarity we divide the proof into three parts. We first consider the symmetric case ρ = 1/2, appealing to a factorization of τ in terms of generalized Rayleigh and Beta random variables which was discovered in [16]. Second, we deal with the spectrally positive case ρ = 1−1/α, with the help of a multiplicative identity in law for τ involving positive stable and shifted Cauchy random variables which was obtained in [11].…”

Section: Introduction and Statement Of The Resultsmentioning

confidence: 99%

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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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“…The symmetric case. Formula (5.12) and Lemma 2.17 in [16] yield together with the normalization (4.1) therein, which is the same as ours, the following independent factorization…”

Section: Proof Of the Theoremsupporting

confidence: 60%

Section: Introduction and Statement Of The Resultsmentioning

confidence: 99%

Unimodality of Hitting Times for Stable Processes

Letemplier

Simon

2014

Lecture Notes in Mathematics

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“…The distribution of τ x plays an important role in various contexts: local times and excursion theory ( [2,7,18]), potential theory ( [3]), penalisation problems ( [30,31]). The estimates of τ x may also prove useful in the study of one-dimensional unimodal Lévy processes, developed recently in [5,6,9].…”

Section: Introduction and Statement Of The Resultsmentioning

confidence: 99%

“…Numerous results are available in this case. In particular, a complete series expansion of the distribution function of τ x is known (see [20] for processes with one-sided jumps, [4,7,21,30] for the symmetric case, and [11] for the general result). Other closely related results for the stable case (unimodality, distributional identities, applications) can be found in [16,26,31].…”

Section: Introduction and Statement Of The Resultsmentioning

confidence: 99%

Hitting times of points for symmetric Lévy processes with completely monotone jumps

Juszczyszyn

Kwaśnicki

2015

Electron. J. Probab.

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Small-space and large-time estimates and asymptotic expansion of the distribution function and (the derivatives of) the density function of hitting times of points for symmetric Lévy processes are studied. The Lévy measure is assumed to have completely monotone density function, and a scaling-type condition inf ξΨ ′′ (ξ)/Ψ ′ (ξ) > 0 is imposed on the Lévy-Khintchine exponent Ψ. Proofs are based on generalised eigenfunction expansion for processes killed upon hitting the origin. R

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“…We remark that the distribution of T 0 has been characterised previously by Yano et al [37] and Cordero [11], using rather different methods; and the Mellin transform above was also obtained, again via the Lamperti transform but without the extended hypergeometric class, in Kuznetsov et al [21].…”

Section: Is a Lévy Process Of The Hypergeometric Class With Parametersmentioning

confidence: 66%

The extended hypergeometric class of Lévy processes

Kyprianou

Pardo

Watson

2014

Journal of Applied Probability

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On the Laws of First Hitting Times of Points for One-Dimensional Symmetric Stable Lévy Processes

Abstract: Several aspects of the laws of first hitting times of points are investigated for one-dimensional symmetric stable Lévy processes. Itô's excursion theory plays a key role in this study.

Cited by 28 publications

References 38 publications

Unimodality of Hitting Times for Stable Processes

Unimodality of Hitting Times for Stable Processes

Hitting times of points for symmetric Lévy processes with completely monotone jumps

The extended hypergeometric class of Lévy processes

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