Maxima of sums of random variables and suprema of stable processes

Bingham, Ν. H.

doi:10.1007/bf00534892

Cited by 91 publications

(97 citation statements)

References 52 publications

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“…It has been proved by Bingham [3], Proposition 3.b and Theorem 4.b, and Rivero [12], section 2.3 that 14) where the constant k is explicitly computed in [3] and is given by:…”

Section: Small Tail Of First Passage Times Of Stable Lévy Processesmentioning

confidence: 99%

Conditioned stable Lévy processes and the Lamperti representation

Caballero

Chaumont

2006

Journal of Applied Probability

136

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By killing a stable Lévy process when it leaves the positive half-line, or by conditioning it to stay positive, or by conditioning it to hit 0 continuously, we obtain three different positive self-similar Markov processes which illustrate the three classes described by Lamperti [10]. For each of these processes, we compute explicitly the infinitesimal generator from which we deduce the characteristics of the underlying Lévy process in the Lamperti representation. The proof of this result bears on the behaviour at time 0 of stable Lévy processes before their first passage time across level 0 which we describe here. As an application, we give the law of the minimum before an independent exponential time of a certain class of Lévy processes. It provides the explicit form of the spacial Wiener-Hopf factor at a particular point and the value of the ruin probability for this class of Lévy processes.

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“…It has been proved by Bingham [3], Proposition 3.b and Theorem 4.b, and Rivero [12], section 2.3 that 14) where the constant k is explicitly computed in [3] and is given by:…”

Section: Small Tail Of First Passage Times Of Stable Lévy Processesmentioning

confidence: 99%

Conditioned stable Lévy processes and the Lamperti representation

Caballero

Chaumont

2006

Journal of Applied Probability

136

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“…In the existing literature, such expressions seem to be available only when X has no positive jumps and the purpose of the present paper is to seek similar expressions when X has no negative jumps. We note that the latter problem dates back to [5], page 282.Our main result (Theorem 1) characterizes the density function f of S 1 as the unique solution to a weakly singular Volterra integral equation of the first kind or, equivalently, as the unique solution to a first order RiemannLiouville fractional differential equation satisfying a boundary condition at …”

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confidence: 96%

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confidence: 99%

The law of the supremum of a stable Lévy process with no negative jumps

Bernyk¹,

Dalang²,

Peškir³

2008

Ann. Probab.

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Let X = (Xt) t≥0 be a stable Lévy process of index α ∈ (1, 2) with no negative jumps and let St = sup 0≤s≤t Xs denote its running supremum for t > 0. We show that the density function ft of St can be characterized as the unique solution to a weakly singular Volterra integral equation of the first kind or, equivalently, as the unique solution to a first-order Riemann-Liouville fractional differential equation satisfying a boundary condition at zero. This yields an explicit series representation for ft. Recalling the familiar relation between St and the first entry time τx of X into [x, ∞), this further translates into an explicit series representation for the density function of τx.1. Introduction. In our study [3] of optimal prediction for a stable Lévy process X = (X t ) t≥0 , we encountered the question of computing the distribution function of S t = sup 0≤s≤t X s for t > 0. In the existing literature, such expressions seem to be available only when X has no positive jumps and the purpose of the present paper is to seek similar expressions when X has no negative jumps. We note that the latter problem dates back to [5], page 282.Our main result (Theorem 1) characterizes the density function f of S 1 as the unique solution to a weakly singular Volterra integral equation of the first kind or, equivalently, as the unique solution to a first order RiemannLiouville fractional differential equation satisfying a boundary condition at

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“…In [8] Ray gives an expression of its density in the symmetric stable case. In [2] Bingham generalises this result for non-spectrally negative stable processes. However, the interest of this part of the work is to point out the role of the scaling property in the obtention of this result.…”

Section: Introductionmentioning

confidence: 61%

“…In [2], Bingham generalizes this result to the case when X is not spectrally negative. Now, we recover this result as a corollary of Theorem 3 and Proposition 6 (see also Lemma 4.1 in [10]).…”

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confidence: 76%

On the scaling property in fluctuation theory for stable Lévy processes

Cordero¹

2010

Teor. Veroyatnost. i Primenen.

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We find an expression for the joint Laplace transform of the law of (T [x,+∞[ , XT [x,+∞[ ) for a Lévy process X, where T [x,+∞[ is the first hitting time of [x, +∞[ by X. When X is an α-stable Lévy process, with 1 < α < 2, we show how to recover from this formula the law of XT [x,+∞[ ; this result was already obtained by D. Ray, in the symmetric case and by N. Bingham, in the case when X is non spectrally negative. Then, we study the behaviour of the time of first passage T [x,+∞[ conditioned to {XT [x,+∞[ − x ≤ h} when h tends to 0. This study brings forward an asymptotic variable T 0 x , which seems to be related to the absolute continuity of the law of the supremum of X.

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Maxima of sums of random variables and suprema of stable processes

Cited by 91 publications

References 52 publications

Conditioned stable Lévy processes and the Lamperti representation

Conditioned stable Lévy processes and the Lamperti representation

The law of the supremum of a stable Lévy process with no negative jumps

On the scaling property in fluctuation theory for stable Lévy processes

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