Tail asymptotics for Shepp-statistics of Brownian motion in $\mathbb {R}^{d}$

Korshunov, Dmitry; Wang, Longmin

doi:10.1007/s10687-019-00357-z

Cited by 10 publications

(11 citation statements)

References 22 publications

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“…Although the probability of multiple simultaneous failures seems very difficult to compute, our first result below, motivated by [15][Thm 1 .1], shows that it can be bounded by the multivariate survival probability p T (u) = P {∀ 1≤i≤d : W i (T ) − c i T > u i } .…”

Section: Introductionmentioning

confidence: 99%

Pandemic-type Failures in Multivariate Brownian Risk Models

Dȩbicki¹,

Hashorva²,

Kriukov³

2020

Preprint

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Modelling of multiple simultaneous failures in insurance, finance and other areas of applied probability is important especially from the point of view of pandemic-type events. A benchmark limiting model for the analysis of multiple failures is the classical d-dimensional Brownian risk model (Brm), see [1]. From both theoretical and practical point of view, of interest is the calculation of the probability of multiple simultaneous failures in a given time horizon. Our main findings here concern the approximation of the probability that at least k out of d components of our Brm fail simultaneously. We derive both sharp bounds and asymptotic approximations of the probability of interest for the finite and the infinite time horizon. Our results extend previous findings of [2,3].

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Section: Introductionmentioning

confidence: 99%

Pandemic-type Failures in Multivariate Brownian Risk Models

Dȩbicki¹,

Hashorva²,

Kriukov³

2020

Preprint

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“…Our proof below is based on the idea of the proof of Korshunov and Wang ( 2020 )[Thm 1.1], where c has zero components, k = d and S = 0 has been considered. Recall the definition of sets and E ( u ) introduced in Eq.…”

Section: Proofsmentioning

confidence: 99%

“…Although the probability of multiple simultaneous failures seems very difficult to compute, our first result below, motivated by Korshunov and Wang ( 2020 )[Thm 1.1], shows that ψ k ( S , T , u ) can be bounded by the multivariate Gaussian survival probability, namely by where When we can approximate p T ( u ) utilising Laplace asymptotic method, see e.g., Korshunov et al ( 2015 ), whereas for small and moderate values of u it can be calculated or simulated with sufficient accuracy. Our next result gives bounds for ψ k ( S , T , u ) in terms of p T ( u ).…”

Section: Introductionmentioning

confidence: 98%

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Pandemic-type failures in multivariate Brownian risk models

2021

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Modelling of multiple simultaneous failures in insurance, finance and other areas of applied probability is important especially from the point of view of pandemic-type events. A benchmark limiting model for the analysis of multiple failures is the classical d-dimensional Brownian risk model (Brm), see Delsing et al. (Methodol. Comput. Appl. Probab. 22(3), 927–948 2020). From both theoretical and practical point of view, of interest is the calculation of the probability of multiple simultaneous failures in a given time horizon. The main findings of this contribution concern the approximation of the probability that at least k out of d components of Brm fail simultaneously. We derive both sharp bounds and asymptotic approximations of the probability of interest for the finite and the infinite time horizon. Our results extend previous findings of Dȩbicki et al. (J. Appl. Probab. 57(2), 597–612 2020) and Dȩbicki et al. (Stoch. Proc. Appl. 128(12), 4171–4206 2018).

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“…The proof of the lower bound is immediate. For the proof of the upper bound we follow the same idea as in the proof of[24][Thm 1.1]. We shall use the standard notation for vectors which are denoted in bold.…”

mentioning

confidence: 99%

Simultaneous Ruin Probability for Two-Dimensional Brownian and Lévy Risk Models

Dȩbicki,

Hashorva,

Michna

2018

Preprint

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The ruin probability in the classical Brownian risk model can be explicitly calculated for both finite and infinite-time horizon. This is not the case for the simultaneous ruin probability in two-dimensional Brownian risk model. Resorting on asymptotic theory, we derive in this contribution approximations of both simultaneous ruin probability and simultaneous ruin time for the two-dimensional Brownian risk model when the initial capital increases to infinity. Given the interest in proportional reinsurance, we consider in some details the case where the correlation is 1. This model is tractable allowing for explicit formulas for the simultaneous ruin probability for linearly dependent spectrally positive Lévy processes. Examples include perturbed Brownian and gamma Lévy processes.

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Tail asymptotics for Shepp-statistics of Brownian motion in $\mathbb {R}^{d}$

Cited by 10 publications

References 22 publications

Pandemic-type Failures in Multivariate Brownian Risk Models

Pandemic-type Failures in Multivariate Brownian Risk Models

Pandemic-type failures in multivariate Brownian risk models

Simultaneous Ruin Probability for Two-Dimensional Brownian and Lévy Risk Models

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