2007
DOI: 10.1007/s11537-007-0662-y
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On the excursion theory for linear diffusions

Abstract: We present a number of important identities related to the excursion theory of linear diffusions. In particular, excursions straddling an independent exponential time are studied in detail. Letting the parameter of the exponential time tend to zero it is seen that these results connect to the corresponding results for excursions of stationary diffusions (in stationary state). We characterize also the laws of the diffusion prior and posterior to the last zero before the exponential time. It is proved using Krei… Show more

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Cited by 47 publications
(44 citation statements)
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“…(2)-mixture. In particular it can be deduced from[18, Equations (46), (49) and (50)] that the density of Y(3) p…”
mentioning
confidence: 99%
“…(2)-mixture. In particular it can be deduced from[18, Equations (46), (49) and (50)] that the density of Y(3) p…”
mentioning
confidence: 99%
“…To calculate the right-hand sides of the above, we consider an excursion of X below u (notice that τ + X (u−) = inf{t > 0 : X t ≥ u} is the first hitting time of X to u): It is known from Salminen et al (2007) and Lemma 2.1 that, n u (e q < ζ(ǫ u ) ∧ T f (ǫ u )) = lim…”
Section: A Proofsmentioning
confidence: 99%
“…For conditioning a transient diffusion to avoid one of the boundaries we refer, for example, to ( [9], [13], [22]). We show in the following Proposition that the dual process X * can be realized as X conditioned in the sense of Doob to exits the segment [l, r] at the endpoints l and r with some specified probabilities.…”
Section: ) a Function I : E → E Is The S-inversion With Fixed Point mentioning
confidence: 99%
“…Then h(l) = 0 and P x 0 -a.s. all trajectories of the process X tend to l and p = P x 0 (H r < H l ) = 0. We follow [22] to define X conditioned to avoid l (i.e. never to hit l in a positive time) as follows.…”
Section: Proposition 5 Assume That X Is Transient and Let H Be Given Bymentioning
confidence: 99%