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.

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.

scholarly journals A Self-Normalized Central Limit Theorem for Markov Random Walks

2012 ◽  
Vol 44 (02) ◽  
pp. 452-478
Author(s):  
Cheng-Der Fuh ◽  
Tian-Xiao Pang
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Random Walks ◽  
State Space ◽  
Central Limit ◽  
Infinite Variance ◽  
Markov Random ◽  
Markov Random Walk

Motivated by the study of the asymptotic normality of the least-squares estimator in the (autoregressive) AR(1) model under possibly infinite variance, in this paper we investigate a self-normalized central limit theorem for Markov random walks. That is, let {X n , n ≥ 0} be a Markov chain on a general state space X with transition probability P and invariant measure π. Suppose that an additive component S n takes values on the real line , and is adjoined to the chain such that {S n , n ≥ 1} is a Markov random walk. Assume that S n = ∑ k=1 n ξ k , and that {ξ n , n ≥ 1} is a nondegenerate and stationary sequence under π that belongs to the domain of attraction of the normal law with zero mean and possibly infinite variance. By making use of an asymptotic variance formula of S n / √n, we prove a self-normalized central limit theorem for S n under some regularity conditions. An essential idea in our proof is to bound the covariance of the Markov random walk via a sequence of weight functions, which plays a crucial role in determining the moment condition and dependence structure of the Markov random walk. As illustrations, we apply our results to the finite-state Markov chain, the AR(1) model, and the linear state space model.

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.

scholarly journals A Self-Normalized Central Limit Theorem for Markov Random Walks

2012 ◽  
Vol 44 (2) ◽  
pp. 452-478 ◽  
Author(s):  
Cheng-Der Fuh ◽  
Tian-Xiao Pang
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Random Walks ◽  
State Space ◽  
Central Limit ◽  
Infinite Variance ◽  
Markov Random ◽  
Markov Random Walk

Motivated by the study of the asymptotic normality of the least-squares estimator in the (autoregressive) AR(1) model under possibly infinite variance, in this paper we investigate a self-normalized central limit theorem for Markov random walks. That is, let {Xn, n ≥ 0} be a Markov chain on a general state space X with transition probability P and invariant measure π. Suppose that an additive component Sn takes values on the real line , and is adjoined to the chain such that {Sn, n ≥ 1} is a Markov random walk. Assume that Sn = ∑k=1nξk, and that {ξn, n ≥ 1} is a nondegenerate and stationary sequence under π that belongs to the domain of attraction of the normal law with zero mean and possibly infinite variance. By making use of an asymptotic variance formula of Sn / √n, we prove a self-normalized central limit theorem for Sn under some regularity conditions. An essential idea in our proof is to bound the covariance of the Markov random walk via a sequence of weight functions, which plays a crucial role in determining the moment condition and dependence structure of the Markov random walk. As illustrations, we apply our results to the finite-state Markov chain, the AR(1) model, and the linear state space model.

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.

Sign in / Sign up

Export Citation Format

Share Document