scholarly journals The Class of Distributions Associated with the Generalized Pollaczek-Khinchine Formula

2012 ◽  
Vol 49 (3) ◽  
pp. 883-887 ◽  
Author(s):  
Offer Kella
Keyword(s):  
Brownian Motion ◽  
Stationary Distribution ◽  
Lévy Process ◽  
Random Variables ◽  
Levy Process ◽  
Independent Random Variables ◽  
Reflected Brownian Motion ◽  
Distribution Of The Maximum ◽  
Workload Distribution ◽  
Stationary Workload

The goal is to identify the class of distributions to which the distribution of the maximum of a Lévy process with no negative jumps and negative mean (equivalently, the stationary distribution of the reflected process) belongs. An explicit new distributional identity is obtained for the case where the Lévy process is an independent sum of a Brownian motion and a general subordinator (nondecreasing Lévy process) in terms of a geometrically distributed sum of independent random variables. This generalizes both the distributional form of the standard Pollaczek-Khinchine formula for the stationary workload distribution in the M/G/1 queue and the exponential stationary distribution of a reflected Brownian motion.

scholarly journals The Class of Distributions Associated with the Generalized Pollaczek-Khinchine Formula

2012 ◽  
Vol 49 (03) ◽  
pp. 883-887 ◽  
Author(s):  
Offer Kella
Keyword(s):  
Brownian Motion ◽  
Stationary Distribution ◽  
Lévy Process ◽  
Random Variables ◽  
Levy Process ◽  
Independent Random Variables ◽  
Reflected Brownian Motion ◽  
Distribution Of The Maximum ◽  
Workload Distribution ◽  
Stationary Workload

The goal is to identify the class of distributions to which the distribution of the maximum of a Lévy process with no negative jumps and negative mean (equivalently, the stationary distribution of the reflected process) belongs. An explicit new distributional identity is obtained for the case where the Lévy process is an independent sum of a Brownian motion and a general subordinator (nondecreasing Lévy process) in terms of a geometrically distributed sum of independent random variables. This generalizes both the distributional form of the standard Pollaczek-Khinchine formula for the stationary workload distribution in the M/G/1 queue and the exponential stationary distribution of a reflected Brownian motion.

scholarly journals A Lévy Process Reflected at a Poisson Age Process

2006 ◽  
Vol 43 (1) ◽  
pp. 221-230 ◽  
Author(s):  
Offer Kella ◽  
Onno Boxma ◽  
Michel Mandjes
Keyword(s):  
Stationary Distribution ◽  
Lévy Process ◽  
Initial Conditions ◽  
Random Variables ◽  
Levy Process ◽  
Independent Random Variables ◽  
Constant Multiple ◽  
The Stability ◽  
The One ◽  
Age Process

We consider a Lévy process with no negative jumps, reflected at a stochastic boundary that is a positive constant multiple of an age process associated with a Poisson process. We show that the stability condition for this process is identical to the one for the case of reflection at the origin. In particular, there exists a unique stationary distribution that is independent of initial conditions. We identify the Laplace-Stieltjes transform of the stationary distribution and observe that it satisfies a decomposition property. In fact, it is a sum of two independent random variables, one of which has the stationary distribution of the process reflected at the origin, and the other the stationary distribution of a certain clearing process. The latter is itself distributed as an infinite sum of independent random variables. Finally, we discuss the tail behavior of the stationary distribution and in particular observe that the second distribution in the decomposition always has a light tail.

scholarly journals A Lévy Process Reflected at a Poisson Age Process

2006 ◽  
Vol 43 (01) ◽  
pp. 221-230 ◽  
Author(s):  
Offer Kella ◽  
Onno Boxma ◽  
Michel Mandjes
Keyword(s):  
Stationary Distribution ◽  
Lévy Process ◽  
Initial Conditions ◽  
Random Variables ◽  
Levy Process ◽  
Independent Random Variables ◽  
Constant Multiple ◽  
The Stability ◽  
The One ◽  
Age Process

We consider a Lévy process with no negative jumps, reflected at a stochastic boundary that is a positive constant multiple of an age process associated with a Poisson process. We show that the stability condition for this process is identical to the one for the case of reflection at the origin. In particular, there exists a unique stationary distribution that is independent of initial conditions. We identify the Laplace-Stieltjes transform of the stationary distribution and observe that it satisfies a decomposition property. In fact, it is a sum of two independent random variables, one of which has the stationary distribution of the process reflected at the origin, and the other the stationary distribution of a certain clearing process. The latter is itself distributed as an infinite sum of independent random variables. Finally, we discuss the tail behavior of the stationary distribution and in particular observe that the second distribution in the decomposition always has a light tail.

Fractional generalizations of filtering problems and their associated fractional Zakai equations

2014 ◽  
Vol 17 (3) ◽  
Author(s):  
Sabir Umarov ◽  
Frederick Daum ◽  
Kenric Nelson
Keyword(s):  
Brownian Motion ◽  
Fractional Derivative ◽  
Lévy Process ◽  
Nonlinear Filtering ◽  
Levy Process ◽  
Zakai Equation ◽  
Filtering Problem

AbstractIn this paper we discuss fractional generalizations of the filtering problem. The ”fractional” nature comes from time-changed state or observation processes, basic ingredients of the filtering problem. The mathematical feature of the fractional filtering problem emerges as the Riemann-Liouville or Caputo-Djrbashian fractional derivative in the associated Zakai equation. We discuss fractional generalizations of the nonlinear filtering problem whose state and observation processes are driven by time-changed Brownian motion or/and Lévy process.

The Distribution of the Maximum of Partial Sums of Independent Random Variables

1950 ◽  
Vol 2 ◽  
pp. 375-384 ◽  
Author(s):  
Mark Kac ◽  
Harry Pollard
Keyword(s):  
Random Variables ◽  
Independent Random Variables ◽  
Partial Sums ◽  
Distribution Of The Maximum

1. The problem. It has been shown [1] that if Xi, + X2, … are independent random variables each of density then(1.1)

scholarly journals On Maxima and Ladder Processes for a Dense Class of Lévy Process

2006 ◽  
Vol 43 (01) ◽  
pp. 208-220 ◽  
Author(s):  
Martijn Pistorius
Keyword(s):  
Laplace Transform ◽  
Error Estimates ◽  
Lévy Process ◽  
Iterative Procedure ◽  
Levy Process ◽  
Distribution Of The Maximum ◽  
Phase Type ◽  
The Laplace Transform ◽  
The Law ◽  
Dense Class

In this paper, we present an iterative procedure to calculate explicitly the Laplace transform of the distribution of the maximum for a Lévy process with positive jumps of phase type. We derive error estimates showing that this iteration converges geometrically fast. Subsequently, we determine the Laplace transform of the law of the upcrossing ladder process and give an explicit pathwise construction of this process.

scholarly journals The Stationary Distribution of Reflected Brownian Motion in a Planar Region

1985 ◽  
Vol 13 (3) ◽  
pp. 744-757 ◽  
Author(s):  
J. M. Harrison ◽  
H. J. Landau ◽  
L. A. Shepp
Keyword(s):  
Brownian Motion ◽  
Stationary Distribution ◽  
Reflected Brownian Motion ◽  
Planar Region

scholarly journals Brownian motion with variable drift: 0-1 laws, hitting probabilities and Hausdorff dimension

2012 ◽  
Vol 153 (2) ◽  
pp. 215-234 ◽  
Author(s):  
YUVAL PERES ◽  
PERLA SOUSI
Keyword(s):  
Brownian Motion ◽  
Hausdorff Dimension ◽  
Lévy Process ◽  
Dirichlet Space ◽  
Levy Process ◽  
Standard Brownian Motion ◽  
Double Points ◽  
Hitting Probabilities

AbstractBy the Cameron–Martin theorem, if a function f is in the Dirichlet space D, then B + f has the same a.s. properties as standard Brownian motion, B. In this paper we examine properties of B + f when fD. We start by establishing a general 0-1 law, which in particular implies that for any fixed f, the Hausdorff dimension of the image and the graph of B + f are constants a.s. (This 0-1 law applies to any Lévy process.) Then we show that if the function f is Hölder(1/2), then B + f is intersection equivalent to B. Moreover, B + f has double points a.s. in dimensions d ≤ 3, while in d ≥ 4 it does not. We also give examples of functions which are Hölder with exponent less than 1/2, that yield double points in dimensions greater than 4. Finally, we show that for d ≥ 2, the Hausdorff dimension of the image of B + f is a.s. at least the maximum of 2 and the dimension of the image of f.

scholarly journals On Maxima and Ladder Processes for a Dense Class of Lévy Process

2006 ◽  
Vol 43 (1) ◽  
pp. 208-220 ◽  
Author(s):  
Martijn Pistorius
Keyword(s):  
Laplace Transform ◽  
Error Estimates ◽  
Lévy Process ◽  
Iterative Procedure ◽  
Levy Process ◽  
Distribution Of The Maximum ◽  
Phase Type ◽  
The Laplace Transform ◽  
The Law ◽  
Dense Class

In this paper, we present an iterative procedure to calculate explicitly the Laplace transform of the distribution of the maximum for a Lévy process with positive jumps of phase type. We derive error estimates showing that this iteration converges geometrically fast. Subsequently, we determine the Laplace transform of the law of the upcrossing ladder process and give an explicit pathwise construction of this process.

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