TY - JOUR

T1 - Predicting the last zero before an exponential time of a spectrally negative Lévy process

AU - Pedraza Ramirez, Jose M.

AU - Baurdoux, Erik

PY - 2023/6/1

Y1 - 2023/6/1

N2 - Given a spectrally negative Lévy process, we predict, in an $L_p$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.

AB - Given a spectrally negative Lévy process, we predict, in an $L_p$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.

KW - Lévy processes

KW - optimal prediction

KW - optimal stopping

UR - http://www.scopus.com/inward/record.url?scp=85159298176&partnerID=8YFLogxK

UR - https://www.mendeley.com/catalogue/d53dcf43-6401-305b-b22b-f835b729cd0f/

U2 - 10.1017/apr.2022.47

DO - 10.1017/apr.2022.47

M3 - Article

SN - 0001-8678

VL - 55

SP - 611

EP - 642

JO - Advances in Applied Probability

JF - Advances in Applied Probability

IS - 2

ER -