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The time of ultimate recovery in Gaussian risk model

  • Krzysztof Dȩbicki
  • Peng LiuEmail author


We analyze the distance \(\mathcal {R}_{T}(u)\) between the first and the last passage time of {X(t) − ct : t ∈ [0, T]} at level u in time horizon T ∈ (0, ], where X is a centered Gaussian process with stationary increments and \(c\in {\mathbb {R}}\), given that the first passage time occurred before T. Under some tractable assumptions on X, we find Δ(u) and G(x) such that
$$\lim\limits_{u\to\infty}\mathbb{P} \left( \mathcal{R}_{T}(u)>{\Delta}(u)x \right)=G(x), $$
for x ≥ 0. We distinguish two scenarios: T < and T = , that lead to qualitatively different asymptotics. The obtained results provide exact asymptotics of the ultimate recovery time after the ruin in Gaussian risk model.


Gaussian risk process Exact asymptotics First ruin time Last ruin time Generalized Pickands-Piterbarg constant 

AMS 2000 Subject Classifications

Primary 60G15 Secondary 60G70 60K25 


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We thank Enkelejd Hashorva for discussions and comments that improved presentation of the results of this contribution. K. Dȩbicki was partially supported by NCN Grant No 2015/17/B/ST1/01102 (2016-2019) whereas P. Liu was supported by the Swiss National Science Foundation Grant 200021-175752/1.


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Authors and Affiliations

  1. 1.Mathematical InstituteUniversity of WrocławWrocławPoland
  2. 2.Department of Actuarial Science, Faculty of Business and EconomicsUniversity of LausanneLausanneSwitzerland

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