On the infimum attained by the reflected fractional Brownian motion
K. Debicki
0
1
K. M. Kosin ski
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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)