Geodesic
On maximal inequalities for Ornstein–Uhlenbeck processes with jumps
Teoriâ veroâtnostej i ee primeneniâ, Tome 66 (2021) no. 4, pp. 895-913 Cet article a éte moissonné depuis la source Math-Net.Ru

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We derive moment and exponential inequalities for the maximum of a generalized Ornstein–Uhlenbeck process under some assumptions on tail distributions of a jump component.
Keywords: generalized Ornstein–Uhlenbeck processes, maximal inequality, the Graversen–Peskir inequality
Mots-clés : the Legendre transformation. 
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N. E. Kordzakhia; A. A. Novikov. On maximal inequalities for Ornstein–Uhlenbeck processes with jumps. Teoriâ veroâtnostej i ee primeneniâ, Tome 66 (2021) no. 4, pp. 895-913. http://geodesic.mathdoc.fr/item/TVP_2021_66_4_a12/

[1] J. Arias-de-Reyna, “On the theorem of Frullani”, Proc. Amer. Math. Soc., 109:1 (1990), 165–175 | DOI | MR | Zbl

[2] O. E. Barndorff-Nielsen, N. Shephard, “Modelling by Lévy processes for financial econometrics”, Lévy processes, Birkhäuser Boston, Boston, MA, 2001, 283–318 | DOI | MR | Zbl

[3] K. Borovkov, A. Novikov, “On a piece-wise deterministic Markov process model”, Statist. Probab. Lett., 53:4 (2001), 421–428 | DOI | MR | Zbl

[4] P. J. Brockwell, R. A. Davis, Yu Yang, “Estimation for nonnegative Lévy-driven CARMA processes”, J. Bus. Econom. Statist., 29:2 (2011), 250–259 | DOI | MR | Zbl

[5] D. L. Burkholder, “Distribution function inequalities for martingales”, Ann. Probability, 1 (1973), 19–42 | DOI | MR | Zbl

[6] M. O. Cáceres, A. A. Budini, “The generalized Ornstein–Uhlenbeck process”, J. Phys. A, 30:24 (1997), 8427–8444 | DOI | MR | Zbl

[7] B. Davis, “On the $L^{p}$ norms of stochastic integrals and other martingales”, Duke Math. J., 43:4 (1976), 697–704 | DOI | MR | Zbl

[8] G. Frullani, “Sopra gli integrali definiti”, Mem. Mat. Fis. Soc. Ital. Sci. Modena, 20 (1828), 44–48

[9] S. E. Graversen, G. Peskir, “Maximal inequalities for the Ornstein–Uhlenbeck process”, Proc. Amer. Math. Soc., 128:10 (2000), 3035–3041 | DOI | MR | Zbl

[10] M. Grigoriu, Applied non-Gaussian processes, Prentice-Hall PTR, Englewood Cliffs, NJ, 1995, xii+442 pp. | Zbl

[11] D. I. Hadjiev, “The first passage problem for generalized Ornstein–Uhlenbeck processes with non-positive jumps”, Séminaire de Probabilités XIX 1983/84, Lecture Notes in Math., 1123, Springer, Berlin, 1985, 80–90 | DOI | MR | Zbl

[12] J. Jacod, A. N. Shiryaev, Limit theorems for stochastic processes, Grundlehren Math. Wiss., 288, Springer-Verlag, Berlin, 1987, xviii+601 pp. | DOI | MR | MR | MR | Zbl | Zbl

[13] Chen Jia, Guohuan Zhao, “Moderate maximal inequalities for the Ornstein–Uhlenbeck process”, Proc. Amer. Math. Soc., 148:8 (2020), 3607–3615 | DOI | MR | Zbl

[14] E. Lenglart, “Relation de domination entre deux processus”, Ann. Inst. H. Poincaré Sect. B (N.S.), 13:2 (1977), 171–179 | MR | Zbl

[15] Ya. A. Lyulko, A. N. Shiryaev, “Sharp maximal inequalities for stochastic processes”, Proc. Steklov Inst. Math., 287:1 (2014), 155–173 | DOI | DOI | MR | Zbl

[16] A. A. Novikov, “On the first passage time of an autoregressive process over a level and an application to a “disorder” problem”, Theory Probab. Appl., 35:2 (1991), 269–279 | DOI | MR | Zbl

[17] A. A. Novikov, “Martingales and first-passage problem for Ornstein–Uhlenbeck processes with a jump component”, Theory Probab. Appl., 48:2 (2004), 288–303 | DOI | DOI | MR | Zbl

[18] A. Novikov, R. E. Melchers, E. Shinjikashvili, N. Kordzakhia, “First passage time of filtered Poisson process with exponential shape function”, Probabilistic Eng. Mech., 20:1 (2005), 57–65 | DOI

[19] A. Osȩkowski, “Sharp maximal inequalities for the martingale square bracket”, Stochastics, 82:6 (2010), 589–605 | DOI | MR | Zbl

[20] D. Revuz, M. Yor, Continuous martingales and Brownian motion, Grundlehren Math. Wiss., 293, 3rd ed., Springer-Verlag, Berlin, 1999, xiv+602 pp. | DOI | MR | Zbl

[21] K. Sato, M. Yamazato, “Stationary processes of Ornstein–Uhlenbeck type”, Probability theory and mathematical statistics (Tbilisi, 1982), Lecture Notes in Math., 1021, Springer, Berlin, 1983, 541–551 | DOI | MR | Zbl

[22] W. Schachermayer, F. Stebegg, “The sharp constant for the Burkholder–Davis–Gundy inequality and non-smooth pasting”, Bernoulli, 24:4A (2018), 2499–2530 | DOI | MR | Zbl

[23] Litan Yan, Bei Zhu, “$L^p$-estimates on diffusion processes”, J. Math. Anal. Appl., 303:2 (2005), 418–435 | DOI | MR | Zbl