Najdanovic, Zorana (2003) Conditioning a Markov chain upon the behaviour of an additive functional. PhD thesis, University of Warwick.

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Abstract

We consider a finite statespace continuous-time irreducible Markov chain (Χt)t≥0 together with some fluctuating additive functional (φt)t≥0. The objective is to condition the Markov process (Xt, φt)t≥0 on the event that the process (φt)t≥0 stays non-negative. There are three possible types of behaviour of the process (φt)t≥0: it can drift to +∞,oscillate, or drift to -∞, and in each of these cases we condition the process (Xt, φt)t≥0 on the event that the process (φt)t≥0 stays non-negative.

In the positive drift case, the event that the process (φt)t≥0 stays non-negative is of positive probability and the process (Xt, φt)t≥0 can be conditioned on it in the standard way. In the oscillating and the negative drift cases, the event that the process (Xt, φt)t≥0 stays non-negative is of zero probability and we cannot condition the process (Xt, φt)t≥0 on it in the standard way. Instead, we look at the limits of laws of the process (Xt, φt)t≥0 conditioned on the event that the process (φt)t≥0 hits large levels before it crosses zero, and of laws of the process (Xt, φt)t≥0 conditioned on the event that the process (φt)t≥0 stays non-negative for a large time. In the oscillating case both limits exists and are equal to the same probability law. In the negative drift case, under certain conditions, both limits exist but give distinct probability laws.

In addition, in the negative drift case, conditioning the process (Xt, φt)t≥0 on the event that the process (φt)t≥0 drifts to +∞ and then further conditioning on the event that the process (φt)t≥0 stays non-negative yields the same result as the limit of conditioning the process (Xt, φt)t≥0 on the event that the process (φt)t≥0 hits large levels before it crosses zero. Similarly, conditioning the process (Xt, φt)t≥0 on the event that the process (φt)t≥0 oscillates and then further conditioning on the event that the process (φt)t≥0 stays non-negative yields the same result as the limit of conditioning the process (Xt, φt)t≥0 on the event that the process (φt)t≥0 stays non-negative for a large time.

Item Type:	Thesis or Dissertation (PhD)
Subjects:	Q Science > QA Mathematics
Library of Congress Subject Headings (LCSH):	Markov processes, Stochastic processes, Functionals
Official Date:	March 2003
Dates:	Date Event March 2003 Submitted
Institution:	University of Warwick
Theses Department:	Department of Statistics
Thesis Type:	PhD
Publication Status:	Unpublished
Supervisor(s)/Advisor:	Jacka, Saul D. ; Warren, Jon (Reader in statistics)
Format of File:	pdf
Extent:	viii, 189 leaves : charts
Language:	eng

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