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Existence of density function for the running maximum of SDEs driven by nontruncated pure-jump Lévy processes
Volume 11, Issue 3 (2024), pp. 303–321
Takuya Nakagawa ORCID icon link to view author Takuya Nakagawa details   Ryoichi Suzuki ORCID icon link to view author Ryoichi Suzuki details  

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https://doi.org/10.15559/24-VMSTA245
Pub. online: 23 January 2024      Type: Research Article      Open accessOpen Access

Received
19 June 2023
Revised
17 November 2023
Accepted
7 January 2024
Published
23 January 2024

Abstract

The existence of density function of the running maximum of a stochastic differential equation (SDE) driven by a Brownian motion and a nontruncated pure-jump process is verified. This is proved by the existence of density function of the running maximum of the Wiener–Poisson functionals resulting from Bismut’s approach to the Malliavin calculus for jump processes.

References

[1] 
Applebaum, D.: Lévy Processes and Stochastic Calculus, 2nd edn. Cambridge Studies in Advanced Mathematics, vol. 116, p. 460. Cambridge University Press, Cambridge (2009). doi: https://doi.org/10.1017/CBO9780511809781. MR2512800
[2] 
Bismut, J.-M.: Calcul des variations stochastique et processus de sauts. Z. Wahrscheinlichkeitstheor. Verw. Geb. 63(2), 147–235 (1983). doi: https://doi.org/10.1007/BF00538963. MR701527
[3] 
Carvajal Pinto, M.B., van Schaik, K.: Optimally stopping at a given distance from the ultimate supremum of a spectrally negative Lévy process. Adv. Appl. Probab. 53(1), 279–299 (2021). doi: https://doi.org/10.1017/apr.2020.54. MR4232757
[4] 
Chaumont, L.: On the law of the supremum of Lévy processes. Ann. Probab. 41(3A), 1191–1217 (2013). doi: https://doi.org/10.1214/11-AOP708. MR3098676
[5] 
Coutin, L., Pontier, M., Ngom, W.: Joint distribution of a Lévy process and its running supremum. J. Appl. Probab. 55(2), 488–512 (2018). doi: https://doi.org/10.1017/jpr.2018.32. MR3832901
[6] 
González Cázares, J.I., Mijatović, A., Uribe Bravo, G.: Geometrically convergent simulation of the extrema of Lévy processes. Math. Oper. Res. 47(2), 1141–1168 (2022). doi: https://doi.org/10.1287/moor.2021.1163. MR4435010
[7] 
Karatzas, I., Shreve, S.E.: Brownian Motion and Stochastic Calculus, 2nd edn. Graduate Texts in Mathematics, vol. 113, p. 470. Springer (1991). doi: https://doi.org/10.1007/978-1-4612-0949-2. MR1121940
[8] 
Komatsu, T.: On the pathwise uniqueness of solutions of one-dimensional stochastic differential equations of jump type. Proc. Jpn. Acad., Ser. A, Math. Sci. 58(8), 353–356 (1982). MR683262
[9] 
Kurtz, T.G.: Random time changes and convergence in distribution under the Meyer-Zheng conditions. Ann. Probab. 19(3), 1010–1034 (1991). MR1112405
[10] 
Kuznetsov, A., Pardo, J.C.: Fluctuations of stable processes and exponential functionals of hypergeometric Lévy processes. Acta Appl. Math. 123, 113–139 (2013). doi: https://doi.org/10.1007/s10440-012-9718-y. MR3010227
[11] 
Nakagawa, T.: ${L^{\alpha -1}}$ distance between two one-dimensional stochastic differential equations driven by a symmetric α-stable process. Jpn. J. Ind. Appl. Math. 37(3), 929–956 (2020). doi: https://doi.org/10.1007/s13160-020-00429-9. MR4142265
[12] 
Protter, P.E.: Stochastic Integration and Differential Equations, 2nd edn. Stochastic Modelling and Applied Probability, vol. 21, p. 419. Springer (2005). doi: https://doi.org/10.1007/978-3-662-10061-5. Corrected third printing. MR2273672
[13] 
Sato, K.: Lévy Processes and Infinitely Divisible Distributions, Revised edn. Cambridge Studies in Advanced Mathematics, vol. 68, p. 521. Cambridge University Press, Cambridge (2013). Translated from the 1990 Japanese original. MR3185174
[14] 
Song, Y., Xie, Y.: Existence of density functions for the running maximum of a Lévy-Itô diffusion. Potential Anal. 48(1), 35–48 (2018). doi: https://doi.org/10.1007/s11118-017-9625-y. MR3745823
[15] 
Song, Y., Zhang, X.: Regularity of density for SDEs driven by degenerate Lévy noises. Electron. J. Probab. 20, 21–27 (2015). doi: https://doi.org/10.1214/EJP.v20-3287. MR3325091
[16] 
Williams, D.: Probability with Martingales. Cambridge Mathematical Textbooks, p. 251. Cambridge University Press, Cambridge (1991). doi: https://doi.org/10.1017/CBO9780511813658. MR1155402

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© 2024 The Author(s). Published by VTeX
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Open access article under the CC BY license.

Keywords
Running maximum density functions Malliavin calculus stochastic differential equations Lévy processes

MSC2010
60H10 60G52 60H07

Funding
The second author was supported by JSPS KAKENHI Grant Number 23K12507.

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