On the infimum attained by the reflected fractional Brownian motion

Extremes, Sep 2014

Let {B H (t):t≥0} be a fractional Brownian motion with Hurst parameter \(H\in (\frac {1}{2},1)\). For the storage process \(Q_{B_{H}}(t)=\sup _{-\infty \le s\le t}\) \(\left (B_{H}(t)-B_{H}(s)-c(t-s)\right )\) we show that, for any T(u)>0 such that \(T(u)=o(u^{\frac {2H-1}{H}})\), $$\mathbb P (\inf_{s\in[0,T(u)]} Q_{B_{H}}(s)>u)\sim\mathbb P(Q_{B_{H}}(0)>u),$$ as \(u\to \infty \). This finding, known in the literature as the strong Piterbarg property, goes in line with previously observed properties of storage processes with self-similar and infinitely divisible input without Gaussian component.

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On the infimum attained by the reflected fractional Brownian motion

K. Debicki 0 1 K. M. Kosin ski 0 1 0 K. M. Kosinski ( ) Institute of Mathematics, University of Warsaw , ul. Banacha 2, 02-097 Warsaw, Poland 1 K. Debicki Mathematical Institute, University of Wrocaw , pl. Grunwaldzki 2/4, 50-384 Wrocaw, Poland Let {BH (t ) : t 0} be a fractional Brownian motion with Hurst parameter H 12 , 1 . For the storage process QBH (t ) sup - QBH (s) > u as u . This finding, known in the literature as the strong Piterbarg property, goes in line with previously observed properties of storage processes with self-similar and infinitely divisible input without Gaussian component. 1 Introduction The analysis of distributional properties of reflected stochastic processes is continuously motivated both by theory- and applied-oriented open problems in probability theory. In this paper we analyze the asymptotic properties of tail distribution of infimum of an important class of such processes, that naturally appear in models of storage (queueing) systems and, by duality to ruin problems, gained broad interest also in problems arising in finance and insurance risk; see, e.g., Norros (2004), Piterbarg (2001), Asmussen (2003), and Asmussen and Albrecher (2010) or a novel work (Hashorva et al. 2013). Consider a fluid queue with infinite buffer capacity, service rate c > 0 and the total inflow by time t modeled by a stochastic process with stationary increments X = {X(t ) : t R}. Following Reich (1958), the stationary storage process that describes the stationary buffer content process, has the following representation QX(t ) = There is a strong motivation for modeling the input process X by a fractional Brownian motion (fBm) BH = {BH (t ) : t R} with H > 1/2, i.e., a centered Gaussian process with stationary increments, continuous sample paths a.s., and variance function B2H (t ) = t 2H . On one hand, such structural properties of fBm as self-similarity and long range dependence, have been statistically confirmed in data analysis of many real traffic processes in modern data-transfer networks. On the other hand, in Taqqu et al. (1997) and Mikosch et al. (2002) it was proven that appropriately scaled aggregation of large number of (integrated) On-Off input processes with regularly varying tail distribution of successive On-times, converges to an fBm with H > 1/2. The importance of fBm storage processes resulted in a vast interest of analysis of the process QBH . In particular finding the properties of finite-dimensional (or at least 1-dimensional) distributions of QBH has been a long standing goal; see Norros (2004) and Piterbarg (2001). The stationarity of increments of BH implies the stationarity of the process QBH , so that, for any fixed t , the random variable QBH (t ) has the same distribution as QBH (0). Nevertheless, apart from the Brownian case H = 21 , the exact distribution of QBH (0) is not known. Therefore, one usually resorts to the exact asymptotics of P(QBH (0) > u), as u . These have been found for the full range of parameter H (0, 1) in Husler and Piterbarg (1999), leading to, P(QBH (0) > u) Au1H Au1H , as u , (1) as u , P(QBH (0) > u), as u . This property is nowadays referred to as the generalized Piterbarg property; see Albin and Samorodnitsky (2004). As a corollary from Eq. 2 one easily gets that for any fixed n > 0 and t1, . . . , tn [0, T ], with u , P min QBH (ti ) > u sup QBH (t) >u 1 1. i=1,...,n t[0,T ] i=1 1 P supt[0,T ] QBH (t) > u This leads to the natural question, whether the minimum over finite number of points can be substituted with the infimum functional, which then leads to P as u . This property shall be referred to as the strong Piterbarg property. The above terminology has been coined by Albin and Samorodnitsky (2004), who, motivated by Piterbarg (2001), considered the case when the input process X belongs to the class of self-similar infinitely divisible stochastic processes with no Gaussian component. They provide general conditions under which Eqs. 2 and 3 hold with QX instead of QBH . The approach in Albin and Samorodnitsky (2004) is based on the assumption that the Levy measure associated with X has heavy tails, which combined with the absence of a Gaussian component allows for more direct and less delicate methods to be employed. It is the light-tailed nature of the Gaussian distribution that renders the problem of the asymptotics of suprema of Gaussian processes hard. Furthermore, infima of Gaussian processes (apart perhaps from the Brownian case) have not been considered systematically. On the high level, the problem stems from the fact that an infimum is, by definition, an intersection of events. If the number of events grows to infinity, then the intersection is much harder to handle than, for instance, the sum of events (which corresponds to the supremum). In this paper we derive exact asymptotics of as u . for the whole range of the parameter H . By comparing them with Eq. 1, he observed a remarkable property th (...truncated)


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K. Dębicki, K. M. Kosiński. On the infimum attained by the reflected fractional Brownian motion, Extremes, 2014, pp. 431-446, Volume 17, Issue 3, DOI: 10.1007/s10687-014-0188-7