Abstract
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 [9] 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 [3], where the same question is studied for a Levy process drifting to $-\infty $. The connection to [3] relies on the so-called Lamperti transformation [15] which links the class of positive self-similar Markov processes with that of Levy processes. Our approach shows that the results in [9] for Bessel processes can also be seen as a consequence of self-similarity.
Highlights
In keeping with the development of a family of prediction problems for Brownian motion and, more generally, Lévy processes, cf. [3, 8, 9, 11] to name but a few, we address the question of predicting the time when a positive self-similar Markov process attains its pathwise global supremum or infimum
A special family of functions associated with unkilled spectrally negative Lévy processes is that of scale functions which are defined as follows
−ξt, −∞, t < e, t ≥ e, where e = inf{t > 0 : ξt = −∞}. It follows that ξis a spectrally negative Lévy process killed at rate q ≥ 0 satisfying limt↑∞ ξt = −∞ whenever q = 0
Summary
Due to the fact that the class of Bessel processes for d > 2 belongs to the class of pssMps with α = 2, it is possible to express the optimal stopping time (1.3) (up to a time-change) in terms of the underlying Lamperti representation ξ (of X) reflected at its infimum This suggests that the simple form of (1.3) in the Bessel case is a consequence of the self-similarity of X and that (1.2). By doing so we extend [3] to the case of pssMps. An additional challenge compared with [3] is that in our setting, the corresponding optimal stopping problem for the underlying Lévy process contains a negative discount factor which requires some careful analysis. This optimal stopping problem is solved in (the self-contained) Section 5
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