The Time at which a Lévy Process Creeps

Griffin, Philip S.; Maller, Ross

doi:10.1214/ejp.v16-945

Cited by 12 publications

(15 citation statements)

References 16 publications

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“…where the asymptotic behaviour of P(Tx ∈ (t, t + ∆]) is given by the RHS of (2) evaluated with x replaced by x − bt. As we will see, (8) and (9) hold in the general case, as does (7). We also have analogous results for the regime where ρ is bounded away from zero and infinity and t → ∞, where we make no assumptions about X other than (H) and that it is a strongly non-lattice subordinator.…”

Section: Introduction and Main Resultssupporting

confidence: 64%

Asymptotic behaviour of first passage time distributions for subordinators

Doney

Rivero

2015

Electron. J. Probab.

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In this paper we establish local estimates for the first passage time of a subordinator under the assumption that it belongs to the Feller class, either at zero or infinity, having as a particular case the subordinators which are in the domain of attraction of a stable distribution, either at zero or infinity. To derive these results we first obtain uniform local estimates for the one dimensional distribution of such a subordinator, which sharpen those obtained by Jain and Pruitt [8]. In the particular case of a subordinator in the domain of attraction of a stable distribution our results are the analogue of the results obtained by the authors in [6] for non-monotone Lévy processes. For subordinators an approach different to that in [6] is necessary because the excursion techniques are not available and also because typically in the non-monotone case the tail distribution of the first passage time has polynomial decrease, while in the subordinator case it is exponential. ‡ Research funded by the CONACYT Project Teoría y aplicaciones de procesos de Lévy where b denotes the drift and Π the Lévy measure of X. We will write ψ * for the exponent of {Xt − bt, t ≥ 0}, so that ψ * (λ) := ψ(λ) − bλ, λ ≥ 0.We are interested in determining the local asymptotic behaviour of the distribution of Tx = inf{t > 0 : Xt > x}. More precisely, we would like to establish estimates for the density function hx(t), (if it exists: it does if b = 0), or more generally of P(Tx ∈ (t, t + ∆]),

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Section: Introduction and Main Resultssupporting

confidence: 64%

Asymptotic behaviour of first passage time distributions for subordinators

Doney

Rivero

2015

Electron. J. Probab.

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“…Now P z (w ′ τ (0) = 0) = 0 if z > 0 so we only need to consider z ≤ 0. If z < 0, then by Lemma 5.1 of [14],…”

Section: General Marginal Convergence Resultsmentioning

confidence: 98%

Path decomposition of ruinous behavior for a general Lévy insurance risk process

Griffin¹,

Maller²

2012

Ann. Appl. Probab.

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We analyze the general Lévy insurance risk process for Lévy measures in the convolution equivalence class S (α) , α > 0, via a new kind of path decomposition. This yields a very general functional limit theorem as the initial reserve level u → ∞, and a host of new results for functionals of interest in insurance risk. Particular emphasis is placed on the time to ruin, which is shown to have a proper limiting distribution, as u → ∞, conditional on ruin occurring under our assumptions. Existing asymptotic results under the S (α) assumption are synthesized and extended, and proofs are much simplified, by comparison with previous methods specific to the convolution equivalence analyses. Additionally, limiting expressions for penalty functions of the type introduced into actuarial mathematics by Gerber and Shiu are derived as straightforward applications of our main results. . This reprint differs from the original in pagination and typographic detail. 1 2 P. S. GRIFFIN AND R. A. MALLER by X. We will refer to this as the general Lévy insurance risk model. It is a generalization of the classical Cramér-Lundberg model, which arises when the claim surplus process is taken to bewhere N t is a Poisson process, U i > 0 form an independent i.i.d. sequence and r > 0. Here r represents the rate of premium inflow and U i the size of the ith claim. The general model allows for income other than through premium inflow and a more realistic claims structure; see Section 2.7.1 of Kyprianou [17]. The assumption X t → −∞ a.s. is a reflection of premiums being set to avoid almost certain ruin for finite u.The primary focus of this paper is on when and how ruin occurs for large reserve levels, that is, as u → ∞. Introduce

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“…Proof. By [15], Corollary 3.1, for z > 0 and γ, r ≥ 0, Thus substituting (7.5), (7.6) and (7.7) into (7.1) gives…”

Section: Finiteness Of ηmentioning

confidence: 93%

“…e αy Π L −1 ,H (dr − s, y + dγ) dy,where the I(γ > 0) term may be omitted in the final expression since the measure there assigns no mass to the set {γ = 0}. On the other hand, if X creeps, that is d H > 0, then from[15], Theorem 3.1(ii),…”

mentioning

confidence: 99%

Convolution equivalent Lévy processes and first passage times

Griffin¹

2013

Ann. Appl. Probab.

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We investigate the behavior of Lévy processes with convolution equivalent Lévy measures, up to the time of first passage over a high level u. Such problems arise naturally in the context of insurance risk where u is the initial reserve. We obtain a precise asymptotic estimate on the probability of first passage occurring by time T . This result is then used to study the process conditioned on first passage by time T . The existence of a limiting process as u → ∞ is demonstrated, which leads to precise estimates for the probability of other events relating to first passage, such as the overshoot. A discussion of these results, as they relate to insurance risk, is also given.

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The Time at which a Lévy Process Creeps

Cited by 12 publications

References 16 publications

Asymptotic behaviour of first passage time distributions for subordinators

Asymptotic behaviour of first passage time distributions for subordinators

Path decomposition of ruinous behavior for a general Lévy insurance risk process

Convolution equivalent Lévy processes and first passage times

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