Lp optimal prediction of the last zero of a spectrally negative Lévy process

Baurdoux, Erik J. ORCID: 0000-0002-5407-0683 and Pedraza, José M. (2024) Lp optimal prediction of the last zero of a spectrally negative Lévy process. Annals of Applied Probability, 34 (1B). 1350 - 1402. ISSN 1050-5164

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Abstract

Given a spectrally negative Lévy process X drifting to infinity, (inspired on the early ideas of Shiryaev (2002)) we are interested in finding a stopping time that minimises the Lp distance (p > 1) with g, the last time X is negative. The solution is substantially more difficult compared to the case p = 1, for which it was shown by Baurdoux and Pedraza (2020) that it is optimal to stop as soon as X exceeds a constant barrier. In the case of p > 1 treated here, we prove that solving this optimal prediction problem is equivalent to solving an optimal stopping problem in terms of a two-dimensional strong Markov process that incorporates the length of the current positive excursion away from 0.We show that an optimal stopping time is now given by the first time that X exceeds a non-increasing and non-negative curve depending on the length of the current positive excursion away from 0. We further characterise the optimal boundary and the value function as the unique solution of a non-linear system of integral equations within a subclass of functions. As examples, the case of a Brownian motion with drift and a Brownian motion with drift perturbed by a Poisson process with exponential jumps are considered.

Item Type:	Article
Official URL:	https://projecteuclid-org.gate3.library.lse.ac.uk/...
Additional Information:	© 2024 Institute of Mathematical Statistics
Divisions:	Statistics
Subjects:	H Social Sciences > HG Finance H Social Sciences > HA Statistics
Date Deposited:	22 Jun 2023 13:54
Last Modified:	14 Sep 2024 09:36
URI:	http://eprints.lse.ac.uk/id/eprint/119468

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