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Conditioning an additive functional of a markov chain to stay nonnegative. II, Hitting a high level

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Jacka, Saul D., Lazic, Zorana and Warren, Jon (2005) Conditioning an additive functional of a markov chain to stay nonnegative. II, Hitting a high level. Advances in Applied Probability, Vol.37 (No.4). pp. 1035-1055. ISSN 0001-8678

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Official URL: http://dx.doi.org/10.1239/aap/1134587752

Abstract

Let (X-t)(t >= 0) be a continuous-time irreducible Markov chain on a finite state space E, let v: E -> R \ {0}, and let (phi(t))(t >= 0) be defined by phi(t) = integral(0)(t) v(X-s) ds. We consider the case in which the process (phi(t))(t >= 0) is oscillating and that in which (phi(t))(t >= 0) has a negative drift. In each of these cases, we condition the process (X-t, phi(t))(t >= 0) on the event that (phi(t))(t >= 0) hits level y before hitting 0 and prove weak convergence of the conditioned process as y -> infinity. In addition, we show the relationship between the conditioning of the process (phi(t))(t >= 0) with a negative drift to oscillate and the conditioning of it to stay nonnegative for a long time, and the relationship between the conditioning of (phi(t))(t >= 0) with a negative drift to drift to infinity and the conditioning of it to hit large levels before hitting 0.

Item Type:	Submitted Journal Article
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science > Statistics
Library of Congress Subject Headings (LCSH):	Markov processes
Journal or Publication Title:	Advances in Applied Probability
Publisher:	Applied Probability Trust
ISSN:	0001-8678
Date:	December 2005
Volume:	Vol.37
Number:	No.4
Number of Pages:	21
Page Range:	pp. 1035-1055
Identification Number:	10.1239/aap/1134587752
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Restricted or Subscription Access
References:	[1] Aczel, J. (1966) Lectures on functional equations and their applications. Academic Press, New York. [2] Bertoin, J. and Doney, R. A. (1994). On conditioning a random walk to stay nonnegative. Ann. Prob. Vol. 22, No. 4, 2152-2167. [3] Jacka, S. D., Lazic, Z., Warren, J. (2005). Conditioning an additive functional of a Markov chain to stay non-negative I: survival for a long time. Submitted to J. Appl. Prob. [4] Najdanovic, Z. (2003). Conditioning a Markov chain upon the behaviour of an additive functional. PhD thesis, University of Warwick. [5] Seneta, E. (1981). Non-negative Matrices and Markov Chains. New York Heidelberg Berlin: Springer-Verlag.
URI:	http://wrap.warwick.ac.uk/id/eprint/34030