Moment and MGF convergence of overshoots and undershoots for Lévy insurance risk processes

Park, Hyun Suk; Maller, Ross

doi:10.1017/s0001867800002767

Cited by 4 publications

(13 citation statements)

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“…A stronger version of (8.2) is in Park and Maller [20]. Since our weaker version is easy to prove, we give a direct proof that does not involve delicate estimation of convolution equivalent integrals as in [20]. Combined with convergence of the overshoot, this weaker result is in fact equivalent to Park and Maller's a priori stronger result on convergence of the mgf of the overshoot.…”

Section: General Marginal Convergence Resultsmentioning

confidence: 81%

Section: General Marginal Convergence Resultsmentioning

confidence: 88%

“…ρ− −ζ(ρ−G Z ρ− )−ηW 0 +λZ ρ− ; W 0 > 0], since λ > 0. Thus (8.11) follows from(8.9).Setting η = ν = ζ = 0 in (8.11) gives lim u→∞ e −λu E (u) e λ(u−X τ (u)− ) = lim u→∞ E (u) e −λX τ (u)− (8.13) = β 2 κ(0, −α) κ(0, λ − α) .This gives a transparent explanation of the mgf result in Theorem 3.3 of[20], and extends it to all λ > 0. Note that letting λ ↓ 0 in the final expression of (8.13), reflects that in the limit, X τ (u)− has mass 1 − β 2 at infinity under P(u) .…”

mentioning

confidence: 86%

“…The special case of (8.23) with ζ = η = 0 is given in Theorem 3.2 of [20]. Results related to (8.23) for the case of a Cramér-Lundberg model with 38 P. S. GRIFFIN AND R. A. MALLER bounded claims density can be found in Corollary 3.2 of Tang and Wei [21].…”

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confidence: 98%

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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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Section: General Marginal Convergence Resultsmentioning

confidence: 81%

Section: General Marginal Convergence Resultsmentioning

confidence: 88%

mentioning

confidence: 86%

mentioning

confidence: 98%

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

Griffin¹,

Maller²

2012

Ann. Appl. Probab.

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“…The results of this paper can also be applied to other fields of applied probability. Recently, Park and Maller (2008) considered the nonlocal asymptotics of the β-order moment of the overshoot and undershoot of a Lévy process with drifts to −∞ almost surely (here β is a positive constant). Since a Lévy process is different from a random walk, in forthcoming papers we will investigate the uniform local asymptotics of a Lévy process and its overshoot and undershoot, and the local asymptotics of the ϕ-moments of the overshoot and undershoot of a Lévy process.…”

Section: Introductionmentioning

confidence: 99%

Asymptotics for the moments of the overshoot and undershoot of a random walk

Cui

Wang²,

Wang³

2009

Advances in Applied Probability

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In this paper we obtain some equivalent conditions and sufficient conditions for the local and nonlocal asymptotics of the φ-moments of the overshoot and undershoot of a random walk, where φ is a nonnegative, long-tailed function. By the strong Markov property, it can be shown that the moments of the overshoot and undershoot and the moments of the first ascending ladder height of a random walk satisfy some renewal equations. Therefore, in this paper we first investigate the local and nonlocal asymptotics for the moments of the first ascending ladder height of a random walk, and then give some equivalent conditions and sufficient conditions for the asymptotics of the solutions to some renewal equations. Using the above results, the main results of this paper are obtained.

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Asymptotic distributions of the overshoot and undershoots for the Lévy insurance risk process in the Cramér and convolution equivalent cases

Griffin

Maller

Schaik

2012

Insurance: Mathematics and Economics

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Recent models of the insurance risk process use a Lévy process to generalise the traditional Cramér-Lundberg compound Poisson model. This paper is concerned with the behaviour of the distributions of the overshoot and undershoots of a high level, for a Lévy process which drifts to −∞ and satisfies a Cramér or a convolution equivalent condition. We derive these asymptotics under minimal conditions in the Cramér case, and compare them with known results for the convolution equivalent case, drawing attention to the striking and unexpected fact that they become identical when certain parameters tend to equality. Thus, at least regarding these quantities, the "medium-heavy" tailed convolution equivalent model segues into the "light-tailed" Cramér model in a natural way. This suggests a usefully expanded flexibility for modelling the insurance risk process. We illustrate this relationship by comparing the asymptotic distributions obtained for the overshoot and undershoots, assuming the Lévy process belongs to the "GTSC" class.

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Moment and MGF convergence of overshoots and undershoots for Lévy insurance risk processes

Cited by 4 publications

References 9 publications

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

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

Asymptotics for the moments of the overshoot and undershoot of a random walk

Asymptotic distributions of the overshoot and undershoots for the Lévy insurance risk process in the Cramér and convolution equivalent cases

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