Predicting the last zero before an exponential time of a spectrally negative Lévy process

Baurdoux, Erik J. ORCID: 0000-0002-5407-0683 and Pedraza, José M. (2023) Predicting the last zero before an exponential time of a spectrally negative Lévy process. Advances in Applied Probability, 55 (2). pp. 611-642. ISSN 0001-8678

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Abstract

Given a spectrally negative Lévy process, we predict, in an $L_1$ sense, the last passage time of the process below zero before an independent exponential time. This optimal prediction problem generalises [2], where the infinite-horizon problem is solved. Using a similar argument as that in [24], we show that this optimal prediction problem is equivalent to solving an optimal prediction problem in a finite-horizon setting. Surprisingly (unlike the infinite-horizon problem), an optimal stopping time is based on a curve that is killed at the moment the mean of the exponential time is reached. That is, an optimal stopping time is the first time the process crosses above a non-negative, continuous, and non-increasing curve depending on time. This curve and the value function are characterised as a solution of a system of nonlinear integral equations which can be understood as a generalisation of the free boundary equations (see e.g. [21, Chapter IV.14.1]) in the presence of jumps. As an example, we numerically calculate this curve in the Brownian motion case and for a compound Poisson process with exponential-sized jumps perturbed by a Brownian motion.

Item Type:	Article
Additional Information:	© The Author(s), 2023.
Divisions:	Statistics
Subjects:	H Social Sciences > HA Statistics Q Science > QA Mathematics
Date Deposited:	31 May 2023 16:39
Last Modified:	25 Apr 2024 17:51
URI:	http://eprints.lse.ac.uk/id/eprint/119290

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