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
Pub. online: 23 January 2024
Type: Research Article
Open Access
Open Access
Received
19 June 2023
19 June 2023
Revised
17 November 2023
17 November 2023
Accepted
7 January 2024
7 January 2024
Published
23 January 2024
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
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
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
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
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
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
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
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
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
Kurtz, T.G.: Random time changes and convergence in distribution under the Meyer-Zheng conditions. Ann. Probab. 19(3), 1010–1034 (1991). MR1112405
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
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
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
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
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
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
Williams, D.: Probability with Martingales. Cambridge Mathematical Textbooks, p. 251. Cambridge University Press, Cambridge (1991). doi: https://doi.org/10.1017/CBO9780511813658. MR1155402
