Gaussian risk models with financial constraints

Dȩbicki, Krzysztof; Hashorva, Enkelejd; Ji, Lanpeng

doi:10.1080/03461238.2013.850442

“…This result is in agreement with, e.g. [2], [14], [20]- [24], [28], [29], and [36], where the approximation of the classical ruin time is considered. The obtained normal approximation of the Parisian ruin time is a new result, even for the Brownian motion risk process with a linear trend.…”

Section: Introductionsupporting

confidence: 90%

Parisian ruin of self-similar Gaussian risk processes

Dȩbicki

¹

,

Hashorva

²

,

Ji

³

2015

Journal of Applied Probability

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“…Therein, the Guassian approximations of both τ * 1 (u) and τ * 2 (u) are derived. First passage times (sometimes called ruin times) are also studied extensively in the framework of insurance risk processes; see the recent articles Griffin and Maller (2012), Griffin (2013), Griffin et al (2013), and Dȩbicki et al (2014), and the monographs Embrechts et al (1997) and Asmussen and Albrecher (2010) for approximations of ruin times of various risk processes. In our framework, τ * 1 (u) can be interpreted as the conditional ruin time of the FBM risk process with tax payments of a loss-carry-forward type.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Approximation of Passage Times of γ-Reflected Processes with FBM Input

Hashorva¹,

Ji²

2014

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Define a γ -reflected process W γ (t) = Y H (t) − γ inf s∈ [0,t] Y H (s), t ≥ 0, with input process {Y H (t), t ≥ 0}, which is a fractional Brownian motion with Hurst index H ∈ (0, 1) and a negative linear trend. In risk theory R γ (u) = u − W γ (t), t ≥ 0, is referred to as the risk process with tax payments of a loss-carry-forward type. For various risk processes, numerous results are known for the approximation of the first and last passage times to 0 (ruin times) when the initial reserve u goes to ∞. In this paper we show that, for the γ -reflected process, the conditional (standardized) first and last passage times are jointly asymptotically Gaussian and completely dependent. An important contribution of this paper is that it links ruin problems with extremes of nonhomogeneous Gaussian random fields defined by Y H , which we also investigate.

show abstract

“…Therein, the Guassian approximations of both τ * 1 (u) and τ * 2 (u) are derived. First passage times (sometimes called ruin times) are also studied extensively in the framework of insurance risk processes; see the recent articles Griffin and Maller (2012), Griffin (2013), Griffin et al (2013), and Dȩbicki et al (2014), and the monographs Embrechts et al (1997) and Asmussen and Albrecher (2010) for approximations of ruin times of various risk processes. In our framework, τ * 1 (u) can be interpreted as the conditional ruin time of the FBM risk process with tax payments of a loss-carry-forward type.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Approximation of Passage Times of γ-Reflected Processes with FBM Input

Hashorva

¹

,

Ji

²

2014

Journal of Applied Probability

Self Cite

View full text Add to dashboard Cite

Define a γ-reflected processWγ(t) =YH(t) - γinfs∈[0,t]YH(s),t≥ 0, with input process {YH(t),t≥ 0}, which is a fractional Brownian motion with Hurst indexH∈ (0, 1) and a negative linear trend. In risk theoryRγ(u) =u-Wγ(t),t≥ 0, is referred to as the risk process with tax payments of a loss-carry-forward type. For various risk processes, numerous results are known for the approximation of the first and last passage times to 0 (ruin times) when the initial reserveugoes to ∞. In this paper we show that, for the γ-reflected process, the conditional (standardized) first and last passage times are jointly asymptotically Gaussian and completely dependent. An important contribution of this paper is that it links ruin problems with extremes of nonhomogeneous Gaussian random fields defined byYH, which we also investigate.

show abstract

Gaussian risk models with financial constraints

Cited by 11 publications

References 23 publications

Parisian ruin of self-similar Gaussian risk processes

Parisian ruin of self-similar Gaussian risk processes

Approximation of Passage Times of γ-Reflected Processes with FBM Input

Approximation of Passage Times of γ-Reflected Processes with FBM Input

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