An occupation time related potential measure for diffusion processes

Abstract

In this paper, for homogeneous diffusion processes, the approach of Y. Li and X. Zhou [Statist. Probab. Lett., 2014, 94: 48–55] is adopted to find expressions of potential measures that are discounted by their joint occupation times over semi-infinite intervals (−∞, a) and (a,∞). The results are expressed in terms of solutions to the differential equations associated with the diffusions generator. Applying these results, we obtain more explicit expressions for Brownian motion with drift, skew Brownian motion, and Brownian motion with two-valued drift, respectively.

Keywords

Laplace transform / occupation time / potential measure / exit time / time-homogeneous diffusion / Brownian motion with two-valued drift / skew Brownian motion

References

[1] AlbrecherH, IvanovsJ, ZhouX. Exit identities for Lévy processes observed at Poisson arrival times. Bernoulli (to appear)

[2] AppuhamillageT, BokilV, ThomannE, WaymireE, WoodB. Occupation and local times for skew Brownian motion with applications to dispersion across an interface. Ann Probab, 2011, 21: 183–214

[3] BorodinA N, SalminenP. Handbook of Brownian Motion-Facts and Formulae. 2nd ed. Basel: Birkh�auser Verlag, 2002

[4] BreuerL.Occupation times for Markov-modulated Brownian motion. J Appl Probab, 2012, 49: 549–565

[5] CaiN, ChenN, WanX. Occupation times of jump-diffusion processes with double exponential jumps and the pricing of options. Math Oper Res, 2010, 35: 412–437

[6] ChenY, YangX, LiY, ZhouX. A joint Laplace transform for pre-exit diffusion of occupation times. Acta Math Sin (Engl Ser) (to appear)

[7] DavydovA, LinetskyV. Structuring, pricing and hedging double-barrier step options. J Comput Finance, 2002, 5: 55–87

[8] DicksonD C M, Egídio dos ReisA D. On the distribution of the duration of negative surplus. Scand Actuar J, 1996, 2: 148–164

[9] DicksonD C M, LiS. The distributions of the time to reach a given level and the duration of negative surplus in the Erlang(2) risk model. Insurance Math Econom, 2013, 52: 490–497

[10] Egídio dos ReisA D. How long is the surplus below zero? Insurance Math Econom, 1993, 12: 23–38

[11] EvansL C. An Introduction to Stochastic Differential Equations.Berkeley: UC Berkeley, 2008

[12] FellerW.Diffusion processes in one dimension.Trans Amer Math Soc, 1954, 77: 1–31

[13] FusaiG. Corridor options and arc-sine law. Ann Appl Probab, 2000, 10: 634–663

[14] GuérinH, RenaudJ F. Joint distribution of a spectrally negative Lévy process and its occupation time, with step option pricing in view. arXiv: 1406.3130

[15] ItôK, McKeanH. Diffusion and Their Sample Paths. New York: Springer-Verlag, 1974

[16] KolkovskaE T, López-MimbelaJ A, MoralesJ V. Occupation measure and local time of classical risk processes. Insurance Math Econom, 2005, 37: 573–584

[17] KyprianouA E, PardoJ C, PérezJ L. Occupation times of refracted Lévy processes. J Theoret Probab, 2014, 27: 1292–1315

[18] LandriaultD, RenaudJ F, ZhouX. Occupation times of spectrally negative Lévy processes with applications. Stochastic Process Appl, 2011, 121: 2629–2641

[19] LandriaultD, ShiT X. Occupation times in the MAP risk model.Insurance Math Econom, 2015, 60: 75–82

[20] LejayA. On the constructions of the skew Brownian motion. Probab Surv, 2006, 3: 413–466

[21] LiB, PalmowskiZ. Fluctuations of Omega-killed spectrally negative Lévy processes. arXiv: 1603.07962

[22] LiB, ZhouX. The joint Laplace transforms for diffusion occupation times. Adv Appl Probab, 2013, 45: 1–19

[23] LiY, WangS, ZhouX, ZhuN. Diffusion occupation time before exiting. Front Math China, 2014, 9: 843–861

[24] LiY, ZhouX. On pre-exit joint occupation times for spectrally negative Lévy processes. Statist Probab Lett, 2014, 94: 48–55

[25] LiY, ZhouX, ZhuN. Two-sided discounted potential measures for spectrally negative Lévy processes. Statist Probab Lett, 2015, 100: 67–76

[26] LinetskyV. Step options. Math Finance, 1999, 9: 55–96

[27] LoeffenR L, RenaudJ F, ZhouX. Occupation times of intervals until first passage times for spectrally negative Lévy processes. Stochastic Process Appl, 2014, 124: 1408–1435

[28] PérezJ L, YamazakiK. On the refracted-reflected spectrally negative Lévy processes. arXiv: 1511.06027

[29] PérezJ L, YamazakiK. Refraction-Reflection Strategies in the Dual Model. arXiv: 1511.07918

[30] PitmanJ, YorM. Laplace transforms related to excursions of a one-dimensional diffusion. Bernoulli, 1999, 5: 249–255

[31] PitmanJ, YorM. Hitting, occupation and inverse local times of one-dimensional diffusions: martingale and excursion approaches. Bernoulli, 2003, 9: 1–24

[32] ProkhorovY V, ShiryaevA N. Probability Theory III.Stochastic Calculus. Berlin: Springer-Verlag, 1998

[33] RenaudJ F. On the time spent in the red by a refracted Lévy process. J Appl Prob, 2014, 51: 1171–1188

[34] YinC, YuenC. Exact joint laws associated with spectrally negative Lévy processes and applications to insurance risk theory. Front Math China, 2014, 9: 1453–1471

RIGHTS & PERMISSIONS

Higher Education Press and Springer-Verlag Berlin Heidelberg