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Corrected Diffusion Approximations for Ruin Probabilities in a Markov Random Walk

Part of: Special processes Limit theorems

Published online by Cambridge University Press:  01 July 2016

C. D. Fuh
C. D. Fuh*
Affiliation:
Academia Sinica
*
∗ Postal address: Institute of Statistical Science, Academia Sinica, Taipei, Taiwan, Republic of China. Research partially supported by NSC 84-2121-M-001-025.
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Abstract

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.

Keywords

DIFFUSION APPROXIMATIONLADDER HEIGHT DISTRIBUTIONMARKOV-DEPENDENT WALD MARTINGALEMARKOV RANDOM WALKGAMBLER'S RUIN

MSC classification

Primary: 60F05: Central limit and other weak theorems
Secondary: 60K15: Markov renewal processes, semi-Markov processes
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 29 , Issue 3 , September 1997 , pp. 695 - 712
Copyright
Copyright © Applied Probability Trust 1997 

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References

Alsmeyer, G. (1994) On the Markov renewal theorem. Stoch. Proc. Appl. 50, 37–56.CrossRefGoogle Scholar
Arndt, K. (1980) Asymptotic properties of the distribution of the supreme of a random walk on a Markov chain. Theory Prob. Appl. 25, 309–323.Google Scholar
Asmussen, S. (1989a) Aspects of matrix Wiener-Hopf factorization in applied probability. Math. Sci. 14, 101–116.Google Scholar
Asmussen, S. (1989b) Risk theory in a Markov environment. Scand. Act. J. 69–100.Google Scholar
Billingsley, P. (1968) Convergence of Probability Measure. Wiley, New York.Google Scholar
Çinlar, E. (1975) Introduction to Stochastic Processes. Prentice-Hall, New Jersey.Google Scholar
Cramer, H. (1930) On the mathematical theory of risk. Skandia Jubilee Volume. Stockholm.Google Scholar
Fuh, C. D. and Lai, T. L. (1995) Ladder variables and moments of first passage time in a Markov random walk. Preprint. Google Scholar
Hoglund, T. (1991) The ruin problem for finite Markov chains. Ann. Prob. 19, 1298–1310.CrossRefGoogle Scholar
Iscoe, I., Ney, P. and Nummelin, E. (1985) Large deviations of uniformly recurrent Markov additive processes. Adv. Appl. Math. 6, 373–412.Google Scholar
Karlin, S. and Dembo, A. (1992) Limit distributions of maximal segmental score among Markov-dependent partial sums. Adv. Appl. Prob. 24, 113–140.Google Scholar
Lalley, S. P. (1984) Limit theorems for first-passage times in linear and non-linear renewal theory. Adv. Appl. Prob. 16, 766–803.Google Scholar
Miller, H. D. (1961) A convexity property in the theory of random variables defined on a finite Markov chain. Ann. Math. Statist. 32, 1261–1270.Google Scholar
Miller, H. D. (1962a) A matrix factorization problem in the theory of random variables defined on a finite Markov chain. Proc. Camb. Phil. Soc. 58, 268–285.Google Scholar
Miller, H. D. (1962b) Absorption probabilities for sums of random variables defined on a finite Markov chain. Proc. Camb. Phil. Soc. 58, 286–298.CrossRefGoogle Scholar
Ney, P. and Nummelin, E. (1987) Markov additive processes. I. Eigenvalue properties and limit theorems. Ann. Prob. 15, 561–592.Google Scholar
Pitman, J. W. (1977) Occupation measure for Markov chains. Adv. Appl. Prob. 9, 69–86.Google Scholar
Schal, M. (1970) Rate of convergence in Markov renewal processes with auxiliary paths. Z. Wahrscheinlichkeitsth. 16, 29–38.Google Scholar
Siegmund, D. (1979) Corrected diffusion approximations in certain random walk problems. Adv. Appl. Prob. 11, 701–719.Google Scholar
Wald, A. (1974) Sequential Analysis. Wiley, New York.Google Scholar