Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-23T05:11:55.707Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Part of: Stochastic processes

Published online by Cambridge University Press:  30 January 2018

Enkelejd Hashorva  and
Lanpeng Ji
Enkelejd Hashorva*
Affiliation:
University of Lausanne
Lanpeng Ji*
Affiliation:
University of Lausanne
*
∗ Postal address: Department of Actuarial Science, University of Lausanne, UNIL-Dorigny 1015 Lausanne, Switzerland.
∗ Postal address: Department of Actuarial Science, University of Lausanne, UNIL-Dorigny 1015 Lausanne, Switzerland.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Define a γ-reflected process Wγ(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 index H ∈ (0, 1) and a negative linear trend. In risk theory Rγ(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 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 YH, which we also investigate.

Keywords

Gaussian approximationpassage timeγ-reflected processworkload processrisk process with taxfractional Brownian motionPiterbarg constantPickands constant

MSC classification

Primary: 60G15: Gaussian processes
Secondary: 60G70: Extreme value theory; extremal processes
Type
Research Article
Information
Journal of Applied Probability , Volume 51 , Issue 3 , September 2014 , pp. 713 - 726
Copyright
© Applied Probability Trust 

References

Adler, R. J. and Taylor, J. E. (2007). Random Fields and Geometry. Springer, New York.Google Scholar
Albin, J. M. P. (1990). On extremal theory for stationary processes. Ann. Prob. 18, 92–128.Google Scholar
Asmussen, S. (1987). Applied Probability and Queues. John Wiley, Chichester.Google Scholar
Asmussen, S. and Albrecher, H. (2010). Ruin Probabilities, 2nd edn. World Scientific, Hackensack, NJ.CrossRefGoogle Scholar
Awad, H. and Glynn, P. (2009). Conditional limit theorems for regulated fractional Brownian motion. Ann. Appl. Prob. 19, 2102–2136.CrossRefGoogle Scholar
Debicki, K. (2002). Ruin probability for Gaussian integrated processes. Stoch. Process. Appl. 98, 151–174.CrossRefGoogle Scholar
Debicki, K. and Mandjes, M. (2003). Exact overflow asymptotics for queues with many Gaussian inputs. J. Appl. Prob. 40, 704–720.CrossRefGoogle Scholar
Debicki, K. and Mandjes, M. (2011). Open problems in Gaussian fluid queueing theory. Queueing Systems 68, 267–273.Google Scholar
Debicki, K. and Tabiś, K. (2011). Extremes of the time-average stationary Gaussian processes. Stoch. Process. Appl. 121, 2049–2063.CrossRefGoogle Scholar
Debicki, K., Hashorva, E. and Ji, L. (2014). Gaussian risk models with financial constraints. Scand. Actuarial J. DOI: 10.1080/03461238.2013.850442.Google Scholar
Debicki, K., Michna, Z. and Rolski, T. (2003). Simulation of the asymptotic constant in some fluid models. Stoch. Models 19, 407–423.CrossRefGoogle Scholar
Dieker, A. B. and Yakir, B. (2014). On asymptotic constants in the theory of extremes for Gaussian processes. Bernoulli 20, 1600–1619.Google Scholar
Duncan, T. E. and Jin, Y. (2008). Maximum queue length of a fluid model with an aggregated fractional Brownian input. In Markov Processes and Related Topics: a Festschrift for Thomas G. Kurtz (Inst. Math. Statist. Collect. 4), Institute of Mathematical Statistics, Beachwood, OH, pp. 235–251.Google Scholar
Embrechts, P., Klüpelberg, C. and Mikosch, T. (1997). Modelling Extremal Events. For Insurance and Finance. Springer, Berlin.Google Scholar
Griffin, P. S. (2013). Convolution equivalent Lévy processes and first passage times. Ann. Appl. Prob. 23, 1506–1543.Google Scholar
Griffin, P. S. and Maller, R. A. (2012). Path decomposition of ruinous behavior for a general Lévy insurance risk process. Ann. Appl. Prob. 22, 1411–1449.Google Scholar
Griffin, P. S., Maller, R. A. and Roberts, D. (2013). Finite time ruin probabilities for tempered stable insurance risk processes. Insurance Math. Econom. 53, 478–489.Google Scholar
Harrison, J. M. (1985). Brownian Motion and Stochastic Flow Systems. John Wiley, New York.Google Scholar
Hashorva, E., Ji, L. and Piterbarg, V. I. (2013). On the supremum of γ-reflected processes with fractional Brownian motion as input. Stoch. Process. Appl. 123, 4111–4127.Google Scholar
Hüsler, J. and Piterbarg, V. (1999). Extremes of a certain class of Gaussian processes. Stoch. Process. Appl. 83, 257–271.Google Scholar
Hüsler, J. and Piterbarg, V. (2008). A limit theorem for the time of ruin in a Gaussian ruin problem. Stoch. Process. Appl. 118, 2014–2021.Google Scholar
Hüsler, J. and Zhang, Y. (2008). On first and last ruin times of Gaussian processes. Statist. Prob. Lett. 78, 1230–1235.CrossRefGoogle Scholar
Kozachenko, Y., Melnikov, A. and Mishura, Y. (2014). On drift parameter estimation in models with fractional Brownian motion. Statistics DOI: 10.1080/02331888.2014.907294.Google Scholar
Mandjes, M. (2007). Large Deviations for Gaussian Queues. John Wiley, Chichester.Google Scholar
Pickands, J. III (1969). Asymptotic properties of the maximum in a stationary Gaussian process. Trans. Amer. Math. Soc. 145, 75–86.Google Scholar
Piterbarg, V. I. (1972). On the paper by J. Pickands Upcrosssing probabilities for stationary Gaussian processes'. Vestnik Moscov. Univ. Ser. I Mat. Meh. 27, 25–30 (in Russian). English translation: Moscow Univ. Math. Bull. 27, 19–23.Google Scholar
Piterbarg, V. I. (1996). Asymptotic Methods in the Theory of Gaussian Processes and Fields (Transl. Math. Monogr. 148) American Mathematical Society, Providence, RI.Google Scholar
Piterbarg, V. I. (2001). Large deviations of a storage process with fractional Browanian motion as input. Extremes 4, 147–164.Google Scholar
Whitt, W. (2002). Stochastic-Process Limits. An Introduction to Stochastic-Process Limits and Their Application to Queues. Springer, New York.CrossRefGoogle Scholar
Zeevi, A. J. and Glynn, P. W. (2000). On the maximum workload of a queue fed by fractional Brownian motion. Ann. Appl. Prob. 10, 1084–1099.Google Scholar