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Register a new userFluctuation theory and exit systems for positive self-similar Markov processes
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For a positive self-similar Markov process, X, we construct a local time for the random set, $Theta$, of times where the process reaches its past supremum. Using this local time we describe an exit system for the excursions of X out of its past supremum. Next, we define and study the ladder process (R,H) associated to a positive self-similar Markov process X, namely a bivariate Markov process with a scaling property whose coordinates are the right inverse of the local time of the random set $Theta$ and the process X sampled on the local time scale. The process (R,H) is described in terms of a ladder process linked to the L{e}vy process associated to X via Lampertis transformation. In the case where X never hits 0, and the upward ladder height process is not arithmetic and has finite mean, we prove the finite-dimensional convergence of (R,H) as the starting point of X tends to 0. Finally, we use these results to provide an alternative proof to the weak convergence of X as the starting point tends to 0. Our approach allows us to address two issues that remained open in Caballero and Chaumont [Ann. Probab. 34 (2006) 1012-1034], namely, how to remove a redundant hypothesis and how to provide a formula for the entrance law of X in the case where the underlying L{e}vy process oscillates.
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This paper addresses the question of predicting when a positive self-similar Markov process X attains its pathwise global supremum or infimum before hitting zero for the first time (if it does at all). This problem has been studied in Glover et al. (2013) under the assumption that X is a positive transient diffusion. We extend their result to the class of positive self-similar Markov processes by establishing a link to Baurdoux and van Schaik (2013), where the same question is studied for a Levy process drifting to minus infinity. The connection to Baurdoux and van Schaik (2013) relies on the so-called Lamperti transformation which links the class of positive self-similar Markov processes with that of Levy processes. Our approach will reveal that the results in Glover et al. (2013) for Bessel processes can also be seen as a consequence of self-similarity.
We prove that a positive self-similar Markov process $(X,mathbb{P})$ that hits 0 in a finite time admits a self-similar recurrent extension that leaves 0 continuously if and only if the underlying L{e}vy process satisfies Cram{e}rs condition.
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In this paper, we start by showing that the intertwining relationship between two minimal Markov semigroups acting on Hilbert spaces implies that any recurrent extensions, in the sense of It^o, of these semigroups satisfy the same intertwining identity. Under mild additional assumptions on the intertwining operator, we prove that the converse also holds. This connection, which relies on the representation of excursion quantities as developed by Fitzsimmons and Getoor, enables us to give an interesting probabilistic interpretation of intertwining relationships between Markov semigroups via excursion theory: two such recurrent extensions that intertwine share, under an appropriate normalization, the same local time at the boundary point. Moreover, in the case when one of the (non-self-adjoint) semigroup intertwines with the one of a quasi-diffusion, we obtain an extension of Kreins theory of strings byshowing that its densely defined spectral measure is absolutely continuous with respect to the measure appearing in the Stieltjes representation of the Laplace exponent of the inverse local time. Finally, we illustrate our results with the class of positive self-similar Markov semigroups and also the reflected generalized Laguerre semigroups. For the latter, we obtain their spectral decomposition and provide, under some conditions, a perturbed spectral gap estimate for its convergence to equilibrium.
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