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

Part of: Stochastic processes Inference from stochastic processes

Published online by Cambridge University Press:  16 January 2023

Erik J. Baurdoux  and
José M. Pedraza
Erik J. Baurdoux*
Affiliation:
London School of Economics and Political Science
José M. Pedraza*
Affiliation:
University of Waterloo
*
*Postal address: Department of Statistics, London School of Economics and Political Science, Houghton Street, London, WC2A 2AE, United Kingdom. Email address: e.j.baurdoux@lse.ac.uk
**Postal address: Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, ON, N2L 3G1, Canada. Email address: josemanuel.pedraza-ramirez@uwaterloo.ca
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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.

Keywords

Lévy processesoptimal predictionoptimal stopping

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory
Secondary: 62M20: Prediction 60G51: Processes with independent increments; L'evy processes
Type
Original Article
Information
Advances in Applied Probability , Volume 55 , Issue 2 , June 2023 , pp. 611 - 642
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Applied Probability Trust

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