Exit identities for diusion processes observed at Poisson arrival times

Abstract

For diffusion processes, we extend various two-sided exit identities to the situation when the process is only observed at arrival times of an independent Poisson process. The results are expressed in terms of solutions to the differential equations associated with the diffusions generators.

Keywords

Time-homogeneous diffusion process / exit problem / Poisson arrival time / Brownian motion

References

[1] Albrecher H, Cheung E C K, Thonhauser S. Randomized observation periods for the compound Poisson risk model dividends. Astin Bull, 2011, 41: 645–672

[2] Albrecher H, Cheung E C K, Thonhauser S. Randomized observation periods for the compound Poisson risk model: the discounted penalty function. Scand Actuar J, 2013, 6: 424–452

[3] Albrecher H, Gerber H U, Shiu E S W. The optimal dividend barrier in the Gamma-Omega model. Eur Actuar J, 2011, 1: 43–55

[4] Albrecher H, Ivanovs J, Zhou X. Exit identities for Lévy processes observed at Poisson arrival times. Bernoulli, 2016, 22(3): 1364–1382

[5] Appuhamillage T, Bokil V, Thomann E, Waymire E, Wood B. Occupation and local times for skew Brownian motion with applications to dispersion across an interface. Ann Probab, 2011, 21: 183–214

[6] Borodin A N, Salminen P. Handbook of Brownian Motion-Facts and Formulae. Berlin: Springer, 2002

[7] Chen Y, Li Y, Zhou X. An occupation time related potential measure for diffusion processes. Front Math China, 2017, 12(3): 559–582

[8] Chen Y, Yang X, Li Y, Zhou X. A joint Laplace transform for pre-exit diffusion occupation times. Acta Math Sin (Engl Ser), 2017, 33(4): 509–525

[9] Feller W. Diffusion processes in one dimension. Trans Amer Math Soc, 1954, 77: 1–31

[10] Kyprianou A E. Introductory Lectures on Fluctuations of Lévy Processes with Applications. Universitext. Berlin: Springer-Verlag, 2006

[11] Lejay A. On the constructions of the skew Brownian motion. Probab Surv, 2006, 3: 413–466

[12] Li Y, Yin C, Zhou X. On the last exit times for spectrally negative Lévy processes. J Appl Probab, 2017, 54: 474–489

[13] Li Y, Zhou X. On pre-exit joint occupation times for spectrally negative Lévy processes. Statist Probab Lett, 2014, 94: 48–55

[14] Li Y, Zhou X, Zhu N. Two-sided discounted potential measures for spectrally negative Lévy processes. Statist Probab Lett, 2015, 100: 67–76

[15] Yin C, Shen Y, Wen Y. Exit problems for jump processes with applications to dividend problems. J Comput Appl Math, 2013, 245: 30{52

[16] Yin C, Yuen K. Exact joint laws associated with spectrally negative Lévy processes and applications to insurance risk theory. Front Math China, 2014, 9(6): 1453–1471

RIGHTS & PERMISSIONS

Higher Education Press