scholarly journals Addendum to Moments of Gamma type and the Brownian supremum process area

2010 ◽  
Vol 7 (0) ◽  
pp. 207-208
Author(s):  
Svante Janson
Keyword(s):  
Supremum Process

scholarly journals The Integral of the Supremum Process of Brownian Motion

2009 ◽  
Vol 46 (2) ◽  
pp. 593-600 ◽  
Author(s):  
Svante Janson ◽  
Niclas Petersson
Keyword(s):  
Brownian Motion ◽  
Laplace Transform ◽  
Explicit Formula ◽  
Local Time ◽  
Time Interval ◽  
Standard Brownian Motion ◽  
Excursion Theory ◽  
Double Laplace Transform ◽  
The Laplace Transform ◽  
Supremum Process

In this paper we study the integral of the supremum process of standard Brownian motion. We present an explicit formula for the moments of the integral (or area)(T) covered by the process in the time interval [0,T]. The Laplace transform of(T) follows as a consequence. The main proof involves a double Laplace transform of(T) and is based on excursion theory and local time for Brownian motion.

scholarly journals Poisson Shot Noise Traffic Model and Approximation of Significant Functionals

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Wissem Jedidi
Keyword(s):  
Shot Noise ◽  
Limit Theorems ◽  
Unified Model ◽  
Traffic Model ◽  
Traffic Modeling ◽  
Input Process ◽  
The Right ◽  
And Control ◽  
Supremum Process ◽  
Poisson Shot Noise

In Internet traffic modeling, many authors presented models based on particular fractal shot noise representations. The inconvenience of these approaches is the multitude of assumptions and the lack of tools to check them. In this paper we propose a unified model based on a general Poisson shot noise representation for the cumulative input process (CIP). We present a procedure of approximation of this process; then we give a procedure for controlling the bandwidth of Internet providers. The approximation and control go via limit theorems for functionals of the CIP, namely, the supremum process, the right inverse, and the storage mapping.

scholarly journals Exit Problems for Spectrally Negative Lévy Processes Reflected at Either the Supremum or the Infimum

2007 ◽  
Vol 44 (4) ◽  
pp. 1012-1030 ◽  
Author(s):  
Xiaowen Zhou
Keyword(s):  
Lévy Processes ◽  
Stable Process ◽  
Levy Processes ◽  
Exit Problem ◽  
Brownian Motion With Drift ◽  
Spectrally Negative Lévy Process ◽  
The Times ◽  
Exit Problems ◽  
Supremum Process ◽  
Spectrally Negative Lévy Processes

For a spectrally negative Lévy process X on the real line, let S denote its supremum process and let I denote its infimum process. For a > 0, let τ(a) and κ(a) denote the times when the reflected processes Ŷ := S − X and Y := X − I first exit level a, respectively; let τ−(a) and κ−(a) denote the times when X first reaches Sτ(a) and Iκ(a), respectively. The main results of this paper concern the distributions of (τ(a), Sτ(a), τ−(a), Ŷτ(a)) and of (κ(a), Iκ(a), κ−(a)). They generalize some recent results on spectrally negative Lévy processes. Our approach relies on results concerning the solution to the two-sided exit problem for X. Such an approach is also adapted to study the excursions for the reflected processes. More explicit expressions are obtained when X is either a Brownian motion with drift or a completely asymmetric stable process.

scholarly journals Exit Problems for Reflected Markov-Modulated Brownian Motion

2012 ◽  
Vol 49 (03) ◽  
pp. 697-709 ◽  
Author(s):  
Lothar Breuer
Keyword(s):  
Brownian Motion ◽  
Queueing Theory ◽  
Joint Distribution ◽  
State Dependent ◽  
Phase Type ◽  
Positive Variation ◽  
Markov Modulated ◽  
Exit Problems ◽  
Supremum Process ◽  
Spectrally Negative Lévy Processes

Let (X, J) denote a Markov-modulated Brownian motion (MMBM) and denote its supremum process by S. For some a > 0, let σ(a) denote the time when the reflected process Y := S - X first surpasses the level a. Furthermore, let σ−(a) denote the last time before σ(a) when X attains its current supremum. In this paper we shall derive the joint distribution of S σ(a), σ−(a), and σ(a), where the latter two will be given in terms of their Laplace transforms. We also provide some remarks on scale matrices for MMBMs with strictly positive variation parameters. This extends recent results for spectrally negative Lévy processes to MMBMs. Due to well-known fluid embedding and state-dependent killing techniques, the analysis applies to Markov additive processes with phase-type jumps as well. The result is of interest to applications such as the dividend problem in insurance mathematics and the buffer overflow problem in queueing theory. Examples will be given for the former.

Weak convergence of first passage time processes

1971 ◽  
Vol 8 (2) ◽  
pp. 417-422 ◽  
Author(s):  
Ward Whitt
Keyword(s):  
Stochastic Process ◽  
Weak Convergence ◽  
First Passage Time ◽  
Continuous Mapping ◽  
Passage Time ◽  
Continuous Functions ◽  
First Passage ◽  
Image Position ◽  
Supremum Process

Let D = D[0, ∞) be the space of all real-valued right-continuous functions on [0, ∞) with limits from the left. For any stochastic process X in D, let the associated supremum process be S(X), wherefor any x ∊ D. It is easy to verify that S: D → D is continuous in any of Skorohod's (1956) topologies extended from D[0,1] to D[0, ∞) (cf. Stone (1963) and Whitt (1970a, c)). Hence, weak convergence Xn ⇒ X in D implies weak convergence S(Xn) ⇒ S(X) in D by virtue of the continuous mapping theorem (cf. Theorem 5.1 of Billingsley (1968)).

scholarly journals Moments of Gamma type and the Brownian supremum process area

2010 ◽  
Vol 7 (0) ◽  
pp. 1-52 ◽  
Author(s):  
Svante Janson
Keyword(s):  
Supremum Process

scholarly journals The Integral of the Supremum Process of Brownian Motion

2009 ◽  
Vol 46 (02) ◽  
pp. 593-600
Author(s):  
Svante Janson ◽  
Niclas Petersson
Keyword(s):  
Brownian Motion ◽  
Laplace Transform ◽  
Explicit Formula ◽  
Local Time ◽  
Time Interval ◽  
Standard Brownian Motion ◽  
Excursion Theory ◽  
Double Laplace Transform ◽  
The Laplace Transform ◽  
Supremum Process

In this paper we study the integral of the supremum process of standard Brownian motion. We present an explicit formula for the moments of the integral (or area)(T) covered by the process in the time interval [0,T]. The Laplace transform of(T) follows as a consequence. The main proof involves a double Laplace transform of(T) and is based on excursion theory and local time for Brownian motion.

scholarly journals Exit Problems for Reflected Markov-Modulated Brownian Motion

2012 ◽  
Vol 49 (3) ◽  
pp. 697-709 ◽  
Author(s):  
Lothar Breuer
Keyword(s):  
Brownian Motion ◽  
Queueing Theory ◽  
Joint Distribution ◽  
State Dependent ◽  
Phase Type ◽  
Positive Variation ◽  
Markov Modulated ◽  
Exit Problems ◽  
Supremum Process ◽  
Spectrally Negative Lévy Processes

Let (X, J) denote a Markov-modulated Brownian motion (MMBM) and denote its supremum process by S. For some a > 0, let σ(a) denote the time when the reflected process Y := S - X first surpasses the level a. Furthermore, let σ−(a) denote the last time before σ(a) when X attains its current supremum. In this paper we shall derive the joint distribution of Sσ(a), σ−(a), and σ(a), where the latter two will be given in terms of their Laplace transforms. We also provide some remarks on scale matrices for MMBMs with strictly positive variation parameters. This extends recent results for spectrally negative Lévy processes to MMBMs. Due to well-known fluid embedding and state-dependent killing techniques, the analysis applies to Markov additive processes with phase-type jumps as well. The result is of interest to applications such as the dividend problem in insurance mathematics and the buffer overflow problem in queueing theory. Examples will be given for the former.

Weak convergence of first passage time processes

1971 ◽  
Vol 8 (02) ◽  
pp. 417-422 ◽  
Author(s):  
Ward Whitt
Keyword(s):  
Stochastic Process ◽  
Weak Convergence ◽  
First Passage Time ◽  
Continuous Mapping ◽  
Passage Time ◽  
Continuous Functions ◽  
First Passage ◽  
Image Position ◽  
Supremum Process

LetD = D[0, ∞) be the space of all real-valued right-continuous functions on [0, ∞) with limits from the left. For any stochastic processXinD,let the associatedsupremum processbeS(X), wherefor anyx ∊ D. It is easy to verify thatS:D→Dis continuous in any of Skorohod's (1956) topologies extended fromD[0,1] toD[0, ∞) (cf. Stone (1963) and Whitt (1970a, c)). Hence, weak convergenceXn⇒XinDimplies weak convergenceS(Xn) ⇒S(X) inDby virtue of the continuous mapping theorem (cf. Theorem 5.1 of Billingsley (1968)).

scholarly journals Exit Problems for Spectrally Negative Lévy Processes Reflected at Either the Supremum or the Infimum

2007 ◽  
Vol 44 (04) ◽  
pp. 1012-1030 ◽  
Author(s):  
Xiaowen Zhou
Keyword(s):  
Lévy Processes ◽  
Stable Process ◽  
Levy Processes ◽  
Exit Problem ◽  
Brownian Motion With Drift ◽  
Spectrally Negative Lévy Process ◽  
The Times ◽  
Exit Problems ◽  
Supremum Process ◽  
Spectrally Negative Lévy Processes

For a spectrally negative Lévy process X on the real line, let S denote its supremum process and let I denote its infimum process. For a > 0, let τ(a) and κ(a) denote the times when the reflected processes Ŷ := S − X and Y := X − I first exit level a, respectively; let τ−(a) and κ−(a) denote the times when X first reaches S τ(a) and I κ(a), respectively. The main results of this paper concern the distributions of (τ(a), S τ(a), τ−(a), Ŷ τ(a)) and of (κ(a), I κ(a), κ−(a)). They generalize some recent results on spectrally negative Lévy processes. Our approach relies on results concerning the solution to the two-sided exit problem for X. Such an approach is also adapted to study the excursions for the reflected processes. More explicit expressions are obtained when X is either a Brownian motion with drift or a completely asymmetric stable process.

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