Corrected diffusion approximations in certain random walk problems

1979 ◽  
Vol 11 (4) ◽  
pp. 701-719 ◽  
Author(s):  
D. Siegmund
Keyword(s):  
Random Walk ◽  
Diffusion Approximation ◽  
Diffusion Approximations ◽  
Infinite Time ◽  
Ladder Height ◽  
Correction Terms

Correction terms are obtained for the diffusion approximation to one- and two-barrier ruin problems in finite and infinite time. The corrections involve moments of ladder-height distributions, and a method is given for calculating them numerically. Examples show that the corrected approximations can be much more accurate than the originals.

Corrected diffusion approximations in certain random walk problems

1979 ◽  
Vol 11 (04) ◽  
pp. 701-719 ◽  
Author(s):  
D. Siegmund
Keyword(s):  
Random Walk ◽  
Diffusion Approximation ◽  
Diffusion Approximations ◽  
Infinite Time ◽  
Ladder Height ◽  
Correction Terms

Correction terms are obtained for the diffusion approximation to one- and two-barrier ruin problems in finite and infinite time. The corrections involve moments of ladder-height distributions, and a method is given for calculating them numerically. Examples show that the corrected approximations can be much more accurate than the originals.

Comment on ‘Corrected diffusion approximations in certain random walk problems'

1986 ◽  
Vol 23 (1) ◽  
pp. 89-96 ◽  
Author(s):  
Michael L. Hogan
Keyword(s):  
Random Walk ◽  
Diffusion Approximation ◽  
Exponential Family ◽  
Ruin Probabilities ◽  
Diffusion Approximations ◽  
Negative Drift ◽  
Family Case ◽  
Correction Terms

Correction terms for the diffusion approximation to the maximum and ruin probabilities for a random walk with small negative drift, obtained by Siegmund (1979) in the exponential family case, are extended by different methods to some non-exponential family cases.

Comment on ‘Corrected diffusion approximations in certain random walk problems'

1986 ◽  
Vol 23 (01) ◽  
pp. 89-96
Author(s):  
Michael L. Hogan
Keyword(s):  
Random Walk ◽  
Diffusion Approximation ◽  
Exponential Family ◽  
Ruin Probabilities ◽  
Diffusion Approximations ◽  
Negative Drift ◽  
Family Case ◽  
Correction Terms

Correction terms for the diffusion approximation to the maximum and ruin probabilities for a random walk with small negative drift, obtained by Siegmund (1979) in the exponential family case, are extended by different methods to some non-exponential family cases.

Corrected Diffusion Approximations for Ruin Probabilities in a Markov Random Walk

1997 ◽  
Vol 29 (03) ◽  
pp. 695-712 ◽  
Author(s):  
C. D. Fuh
Keyword(s):  
Random Walk ◽  
State Space ◽  
Stopping Times ◽  
Ruin Probabilities ◽  
Diffusion Approximations ◽  
Markov Random ◽  
Markov Random Walk ◽  
Finite State ◽  
Correction Terms ◽  
Finite State Space

Let (X, S) = {(Xn , Sn ); n ≧0} be a Markov random walk with finite state space. For a ≦ 0 < b define the stopping times τ= inf {n:Sn > b} and T= inf{n:Sn ∉(a, b)}. The diffusion approximations of a one-barrier probability P {τ < ∝ | X o = i}, and a two-barrier probability P{ST ≧b | X o = i} with correction terms are derived. Furthermore, to approximate the above ruin probabilities, the limiting distributions of overshoot for a driftless Markov random walk are involved.

Corrected Diffusion Approximations for Ruin Probabilities in a Markov Random Walk

1997 ◽  
Vol 29 (3) ◽  
pp. 695-712 ◽  
Author(s):  
C. D. Fuh
Keyword(s):  
Random Walk ◽  
State Space ◽  
Stopping Times ◽  
Ruin Probabilities ◽  
Diffusion Approximations ◽  
Markov Random ◽  
Markov Random Walk ◽  
Finite State ◽  
Correction Terms ◽  
Finite State Space

Let (X, S) = {(Xn, Sn); n ≧0} be a Markov random walk with finite state space. For a ≦ 0 < b define the stopping times τ= inf {n:Sn > b} and T= inf{n:Sn∉(a, b)}. The diffusion approximations of a one-barrier probability P {τ < ∝ | Xo= i}, and a two-barrier probability P{ST ≧b | Xo = i} with correction terms are derived. Furthermore, to approximate the above ruin probabilities, the limiting distributions of overshoot for a driftless Markov random walk are involved.

scholarly journals Uniform renewal theory with applications to expansions of random geometric sums

2007 ◽  
Vol 39 (4) ◽  
pp. 1070-1097 ◽  
Author(s):  
J. Blanchet ◽  
P. Glynn
Keyword(s):  
Asymptotic Expansion ◽  
Asymptotic Expansions ◽  
Random Variables ◽  
Renewal Theory ◽  
Random Variable ◽  
Higher Order ◽  
Order Correction ◽  
Diffusion Approximations ◽  
Geometric Sums ◽  
Correction Terms

Consider a sequence X = (Xn: n ≥ 1) of independent and identically distributed random variables, and an independent geometrically distributed random variable M with parameter p. The random variable SM = X1 + ∙ ∙ ∙ + XM is called a geometric sum. In this paper we obtain asymptotic expansions for the distribution of SM as p ↘ 0. If EX1 > 0, the asymptotic expansion is developed in powers of p and it provides higher-order correction terms to Renyi's theorem, which states that P(pSM > x) ≈ exp(-x/EX1). Conversely, if EX1 = 0 then the expansion is given in powers of √p. We apply the results to obtain corrected diffusion approximations for the M/G/1 queue. These expansions follow in a unified way as a consequence of new uniform renewal theory results that are also developed in this paper.

On some time-non-homogeneous diffusion approximations to queueing systems

1987 ◽  
Vol 19 (4) ◽  
pp. 974-994 ◽  
Author(s):  
V. Giorno ◽  
A. G. Nobile ◽  
L. M. Ricciardi
Keyword(s):  
Diffusion Approximation ◽  
Busy Period ◽  
Queueing Systems ◽  
Time Dependent ◽  
Single Server ◽  
Time Varying ◽  
Diffusion Approximations ◽  
Linear Drift ◽  
Transition Functions ◽  
Fcfs Discipline

Time-non-homogeneous diffusion approximations to single server–single queue–FCFS discipline systems are considered. Under various assumptions on the nature of the time-dependent functions appearing in the infinitesimal moments the transient and the regime behaviour of the approximating diffusions are analysed in some detail. Special attention is then given to the study of a diffusion approximation characterized by a linear drift and by a periodically time-varying infinitesimal variance. Unlike the behaviour of transition functions and moments, the p.d.f. of the busy period is seen to be unaffected by the presence of such periodicity.

scholarly journals Uniform renewal theory with applications to expansions of random geometric sums

2007 ◽  
Vol 39 (04) ◽  
pp. 1070-1097 ◽  
Author(s):  
J. Blanchet ◽  
P. Glynn
Keyword(s):  
Asymptotic Expansion ◽  
Asymptotic Expansions ◽  
Random Variables ◽  
Renewal Theory ◽  
Random Variable ◽  
Higher Order ◽  
Order Correction ◽  
Diffusion Approximations ◽  
Geometric Sums ◽  
Correction Terms

Consider a sequenceX= (Xn:n≥ 1) of independent and identically distributed random variables, and an independent geometrically distributed random variableMwith parameterp. The random variableSM=X1+ ∙ ∙ ∙ +XMis called a geometric sum. In this paper we obtain asymptotic expansions for the distribution ofSMasp↘ 0. If EX1&gt; 0, the asymptotic expansion is developed in powers ofpand it provides higher-order correction terms to Renyi's theorem, which states that P(pSM&gt;x) ≈ exp(-x/EX1). Conversely, if EX1= 0 then the expansion is given in powers of √p. We apply the results to obtain corrected diffusion approximations for the M/G/1 queue. These expansions follow in a unified way as a consequence of new uniform renewal theory results that are also developed in this paper.

Understanding the Wiener–Hopf factorization for the simple random walk

1994 ◽  
Vol 31 (2) ◽  
pp. 561-563 ◽  
Author(s):  
Joanne Kennedy
Keyword(s):  
Random Walk ◽  
Sample Path ◽  
Direct Proof ◽  
Simple Random Walk ◽  
Ladder Height

We give a sample path proof of the well-known Wiener–Hopf identity F = G– + G+ – G– ∗G+ which relates the ladder-height distributions G– and G+ of a simple random walk to the step distribution F. Unlike previous approaches this direct proof is both simple and intuitive.

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