Journal of Applied Probability

Approximation of passage times of γ-reflected processes with FBM input

Enkelejd Hashorva and Lanpeng Ji

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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.

Article information

J. Appl. Probab., Volume 51, Number 3 (2014), 713-726.

First available in Project Euclid: 5 September 2014

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G15: Gaussian processes
Secondary: 60G70: Extreme value theory; extremal processes

Gaussian approximation passage time γ-reflected process workload process risk process with tax fractional Brownian motion Piterbarg constant Pickands constant


Hashorva, Enkelejd; Ji, Lanpeng. Approximation of passage times of γ-reflected processes with FBM input. J. Appl. Probab. 51 (2014), no. 3, 713--726. doi:10.1239/jap/1409932669.

Export citation


  • Adler, R. J. and Taylor, J. E. (2007). Random Fields and Geometry. Springer, New York.
  • Albin, J. M. P. (1990). On extremal theory for stationary processes. Ann. Prob. 18, 92–128.
  • Asmussen, S. (1987). Applied Probability and Queues. John Wiley, Chichester.
  • Asmussen, S. and Albrecher, H. (2010). Ruin Probabilities, 2nd edn. World Scientific, Hackensack, NJ.
  • Awad, H. and Glynn, P. (2009). Conditional limit theorems for regulated fractional Brownian motion. Ann. Appl. Prob. 19, 2102–2136.
  • Dębicki, K. (2002). Ruin probability for Gaussian integrated processes. Stoch. Process. Appl. 98, 151–174.
  • Dębicki, K. and Mandjes, M. (2003). Exact overflow asymptotics for queues with many Gaussian inputs. J. Appl. Prob. 40, 704–720.
  • Dębicki, K. and Mandjes, M. (2011). Open problems in Gaussian fluid queueing theory. Queueing Systems 68, 267–273.
  • Dębicki, K. and Tabiś, K. (2011). Extremes of the time-average stationary Gaussian processes. Stoch. Process. Appl. 121, 2049–2063.
  • Dębicki, K., Hashorva, E. and Ji, L. (2014). Gaussian risk models with financial constraints. Scand. Actuarial J. DOI: 10.1080/03461238.2013.850442.
  • Dębicki, K., Michna, Z. and Rolski, T. (2003). Simulation of the asymptotic constant in some fluid models. Stoch. Models 19, 407–423.
  • Dieker, A. B. and Yakir, B. (2014). On asymptotic constants in the theory of extremes for Gaussian processes. Bernoulli 20, 1600–1619.
  • 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.
  • Embrechts, P., Klüpelberg, C. and Mikosch, T. (1997). Modelling Extremal Events. For Insurance and Finance. Springer, Berlin.
  • Griffin, P. S. (2013). Convolution equivalent Lévy processes and first passage times. Ann. Appl. Prob. 23, 1506–1543.
  • 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.
  • 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.
  • Harrison, J. M. (1985). Brownian Motion and Stochastic Flow Systems. John Wiley, New York.
  • Hashorva, E., Ji, L. and Piterbarg, V. I. (2013). On the supremum of $\gamma$-reflected processes with fractional Brownian motion as input. Stoch. Process. Appl. 123, 4111–4127.
  • Hüsler, J. and Piterbarg, V. (1999). Extremes of a certain class of Gaussian processes. Stoch. Process. Appl. 83, 257–271.
  • 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.
  • Hüsler, J. and Zhang, Y. (2008). On first and last ruin times of Gaussian processes. Statist. Prob. Lett. 78, 1230–1235.
  • 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.
  • Mandjes, M. (2007). Large Deviations for Gaussian Queues. John Wiley, Chichester.
  • Pickands, J., III (1969). Asymptotic properties of the maximum in a stationary Gaussian process. Trans. Amer. Math. Soc. 145, 75–86.
  • 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.
  • Piterbarg, V. I. (1996). Asymptotic Methods in the Theory of Gaussian Processes and Fields (Transl. Math. Monogr. 148) American Mathematical Society, Providence, RI.
  • Piterbarg, V. I. (2001). Large deviations of a storage process with fractional Browanian motion as input. Extremes 4, 147–164.
  • Whitt, W. (2002). Stochastic-Process Limits. An Introduction to Stochastic-Process Limits and Their Application to Queues. Springer, New York.
  • 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. \endharvreferences