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

Erik J. Baurdoux and José M. Pedraza

LSE Research Online Documents on Economics from London School of Economics and Political Science, LSE Library

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.

Keywords: Lévy processes; optimal prediction; optimal stopping (search for similar items in EconPapers)
JEL-codes: C1 (search for similar items in EconPapers)
Pages: 32 pages
Date: 2023-06-01
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Published in Advances in Applied Probability, 1, June, 2023, 55(2), pp. 611 - 642. ISSN: 0001-8678

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