Distributions of stopping times for compound poisson processes with positive jumps and linear boundaries

S. Zacks; D. Perry; D. Bshouty; S. Bar-Lev

doi:10.1080/15326349908807527

Distributions of stopping times for compound poisson processes with positive jumps and linear boundaries

S. Zacks, D. Perry, D. Bshouty, S. Bar-Lev

Department of Statistics

Research output: Contribution to journal › Article › peer-review

Abstract

Let {Y_t, t ≥ 0} be a compound Poisson process, i.e., Y_t = ∑_i=0^NtX_i, X₀ = 0, X₁, X₂, . . . i.i.d. positive random variables and {N_t, t ≥ 0} a time homogeneous Poisson jump process. We consider two linear boundaries, b_L (t) = -β₁ + 1 and b_U (t) = β₂ + t, β₁, β₂ > 0 and the stopping times T_L = inf{t : Y_t = b_L(t)}, T_U = inf{t : Y_t ≥ b_U(t)} and T = min{T_L,T_U}. Laplace Stieltjes transforms are developed for the distributions of T_U and T for general distribution of X_i (i ≥ 1). These transforms are obtained by analyzing the sample path behavior of the process {Y_t}. The results are applicable to reliability theory, sequential analysis, queuing theory, dam theory, risk analysis, and more.

Original language	English
Pages (from-to)	89-101
Number of pages	13
Journal	Communications in Statistics. Part C: Stochastic Models
Volume	15
Issue number	1
DOIs	https://doi.org/10.1080/15326349908807527
State	Published - 1999

Keywords

Laplace transforms
Linear boundaries
Martingales
Queuing theory
Stopping times

ASJC Scopus subject areas

Modeling and Simulation

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10.1080/15326349908807527

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@article{78c80f2d839d4d32b47c9357a2500721,

title = "Distributions of stopping times for compound poisson processes with positive jumps and linear boundaries",

abstract = "Let \{Yt, t ≥ 0\} be a compound Poisson process, i.e., Yt = ∑i=0NtXi, X0 = 0, X1, X2, . . . i.i.d. positive random variables and \{Nt, t ≥ 0\} a time homogeneous Poisson jump process. We consider two linear boundaries, bL (t) = -β1 + 1 and bU (t) = β2 + t, β1, β2 > 0 and the stopping times TL = inf\{t : Yt = bL(t)\}, TU = inf\{t : Yt ≥ bU(t)\} and T = min\{TL,TU\}. Laplace Stieltjes transforms are developed for the distributions of TU and T for general distribution of Xi (i ≥ 1). These transforms are obtained by analyzing the sample path behavior of the process \{Yt\}. The results are applicable to reliability theory, sequential analysis, queuing theory, dam theory, risk analysis, and more.",

keywords = "Laplace transforms, Linear boundaries, Martingales, Queuing theory, Stopping times",

author = "S. Zacks and D. Perry and D. Bshouty and S. Bar-Lev",

year = "1999",

doi = "10.1080/15326349908807527",

language = "English",

volume = "15",

pages = "89--101",

journal = "Communications in Statistics. Part C: Stochastic Models",

issn = "0882-0287",

publisher = "Taylor and Francis",

number = "1",

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TY - JOUR

T1 - Distributions of stopping times for compound poisson processes with positive jumps and linear boundaries

AU - Zacks, S.

AU - Perry, D.

AU - Bshouty, D.

AU - Bar-Lev, S.

PY - 1999

Y1 - 1999

N2 - Let {Yt, t ≥ 0} be a compound Poisson process, i.e., Yt = ∑i=0NtXi, X0 = 0, X1, X2, . . . i.i.d. positive random variables and {Nt, t ≥ 0} a time homogeneous Poisson jump process. We consider two linear boundaries, bL (t) = -β1 + 1 and bU (t) = β2 + t, β1, β2 > 0 and the stopping times TL = inf{t : Yt = bL(t)}, TU = inf{t : Yt ≥ bU(t)} and T = min{TL,TU}. Laplace Stieltjes transforms are developed for the distributions of TU and T for general distribution of Xi (i ≥ 1). These transforms are obtained by analyzing the sample path behavior of the process {Yt}. The results are applicable to reliability theory, sequential analysis, queuing theory, dam theory, risk analysis, and more.

AB - Let {Yt, t ≥ 0} be a compound Poisson process, i.e., Yt = ∑i=0NtXi, X0 = 0, X1, X2, . . . i.i.d. positive random variables and {Nt, t ≥ 0} a time homogeneous Poisson jump process. We consider two linear boundaries, bL (t) = -β1 + 1 and bU (t) = β2 + t, β1, β2 > 0 and the stopping times TL = inf{t : Yt = bL(t)}, TU = inf{t : Yt ≥ bU(t)} and T = min{TL,TU}. Laplace Stieltjes transforms are developed for the distributions of TU and T for general distribution of Xi (i ≥ 1). These transforms are obtained by analyzing the sample path behavior of the process {Yt}. The results are applicable to reliability theory, sequential analysis, queuing theory, dam theory, risk analysis, and more.

KW - Laplace transforms

KW - Linear boundaries

KW - Martingales

KW - Queuing theory

KW - Stopping times

UR - https://www.scopus.com/pages/publications/0001935954

U2 - 10.1080/15326349908807527

DO - 10.1080/15326349908807527

M3 - Article

AN - SCOPUS:0001935954

SN - 0882-0287

VL - 15

SP - 89

EP - 101

JO - Communications in Statistics. Part C: Stochastic Models

JF - Communications in Statistics. Part C: Stochastic Models

IS - 1

ER -

Distributions of stopping times for compound poisson processes with positive jumps and linear boundaries

Abstract

Keywords

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