Gaussian Volterra processes: Asymptotic growth and statistical estimation
Authors:
Yuliya Mishura, Kostiantyn Ralchenko and Sergiy Shklyar
Journal:
Theor. Probability and Math. Statist. 108 (2023), 149-167
MSC (2020):
Primary 60G22, 60G15, 60G17, 60G18, 62F12
DOI:
https://doi.org/10.1090/tpms/1190
Published electronically:
May 2, 2023
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Additional Information
Abstract: The paper is devoted to three-parametric self-similar Gaussian Volterra processes that generalize fractional Brownian motion. We study the asymptotic growth of such processes and the properties of long- and short-range dependence. Then we consider the problem of the drift parameter estimation for Ornstein–Uhlenbeck process driven by Gaussian Volterra process under consideration. We construct a strongly consistent estimator and investigate its asymptotic properties. Namely, we prove that it has the Cauchy asymptotic distribution.
References
- George E. Andrews, Richard Askey, and Ranjan Roy, Special functions, Encyclopedia of Mathematics and its Applications, vol. 71, Cambridge University Press, Cambridge, 1999. MR 1688958, DOI 10.1017/CBO9781107325937
- Antoine Ayache and Jacques Levy Vehel, The generalized multifractional Brownian motion, Stat. Inference Stoch. Process. 3 (2000), no. 1-2, 7–18. 19th “Rencontres Franco-Belges de Statisticiens” (Marseille, 1998). MR 1819282, DOI 10.1023/A:1009901714819
- R. Belfadli, K. Es-Sebaiy, and Y. Ouknine, Parameter estimation for fractional Ornstein–Uhlenbeck processes: non-ergodic case, Frontiers in Science and Engineering 1 (2011), 1–16.
- Konstantin Borovkov, Yuliya Mishura, Alexander Novikov, and Mikhail Zhitlukhin, Bounds for expected maxima of Gaussian processes and their discrete approximations, Stochastics 89 (2017), no. 1, 21–37. MR 3574693, DOI 10.1080/17442508.2015.1126282
- Mohamed El Machkouri, Khalifa Es-Sebaiy, and Youssef Ouknine, Least squares estimator for non-ergodic Ornstein-Uhlenbeck processes driven by Gaussian processes, J. Korean Statist. Soc. 45 (2016), no. 3, 329–341. MR 3527650, DOI 10.1016/j.jkss.2015.12.001
- Paul Embrechts and Makoto Maejima, Selfsimilar processes, Princeton Series in Applied Mathematics, Princeton University Press, Princeton, NJ, 2002. MR 1920153
- John Lamperti, Semi-stable stochastic processes, Trans. Amer. Math. Soc. 104 (1962), 62–78. MR 138128, DOI 10.1090/S0002-9947-1962-0138128-7
- M. B. Marcus, Upper bounds for the asymptotic maxima of continuous Gaussian processes, Ann. Math. Statist. 43 (1972), 522–533. MR 388519, DOI 10.1214/aoms/1177692633
- Y. Mishura, G. Shevchenko, and S. Shklyar, Gaussian processes with Volterra kernels, Stochastic Processes, Statistical Methods, and Engineering Mathematics. SPAS 2019 (S. Silvestrov, A. Malyarenko, Y. Ni, and M. Rančić, eds.), Springer, Cham, 2022, pp. 249–276.
- Yuliya Mishura and Sergiy Shklyar, Gaussian Volterra processes with power-type kernels. Part I, Mod. Stoch. Theory Appl. 9 (2022), no. 3, 313–338. MR 4462026, DOI 10.15559/22-VMSTA205
- Yuliya Mishura and Sergiy Shklyar, Gaussian Volterra processes with power-type kernels. Part II, Mod. Stoch. Theory Appl. 9 (2022), no. 4, 431–452. MR 4510382, DOI 10.15559/22-VMSTA211
- Ilkka Norros, Esko Valkeila, and Jorma Virtamo, An elementary approach to a Girsanov formula and other analytical results on fractional Brownian motions, Bernoulli 5 (1999), no. 4, 571–587. MR 1704556, DOI 10.2307/3318691
- K. V. Ral′chenko, Approximation of multifractional Brownian motion by absolutely continuous processes, Teor. Ĭmovīr. Mat. Stat. 82 (2010), 115–127 (Ukrainian, with English, Russian and Ukrainian summaries); English transl., Theory Probab. Math. Statist. 82 (2011), 115–127. MR 2790487, DOI 10.1090/S0094-9000-2011-00831-9
- Tommi Sottinen and Lauri Viitasaari, Stochastic analysis of Gaussian processes via Fredholm representation, Int. J. Stoch. Anal. , posted on (2016), Art. ID 8694365, 15. MR 3536393, DOI 10.1155/2016/8694365
- Adil Yazigi, Representation of self-similar Gaussian processes, Statist. Probab. Lett. 99 (2015), 94–100. MR 3321501, DOI 10.1016/j.spl.2015.01.012
References
- G. E. Andrews, R. Askey, and R. Roy, Special functions, Encyclopedia of Mathematics and its Applications, vol. 71, Cambridge University Press, Cambridge, 1999. MR 1688958
- A. Ayache and J. Levy Vehel, The generalized multifractional Brownian motion, Stat. Inference Stoch. Process. 3 (2000), no. 1–2, 7–18. MR 1819282
- R. Belfadli, K. Es-Sebaiy, and Y. Ouknine, Parameter estimation for fractional Ornstein–Uhlenbeck processes: non-ergodic case, Frontiers in Science and Engineering 1 (2011), 1–16.
- K. Borovkov, Y. Mishura, A. Novikov, and M. Zhitlukhin, Bounds for expected maxima of Gaussian processes and their discrete approximations, Stochastics 89 (2017), no. 1, 21–37. MR 3574693
- M. El Machkouri, K. Es-Sebaiy, and Y. Ouknine, Least squares estimator for non-ergodic Ornstein–Uhlenbeck processes driven by Gaussian processes, J. Korean Statist. Soc. 45 (2016), no. 3, 329–341. MR 3527650
- P. Embrechts and M. Maejima, Selfsimilar processes, Princeton Series in Applied Mathematics, Princeton University Press, Princeton, NJ, 2002. MR 1920153
- J. Lamperti, Semi-stable stochastic processes, Trans. Amer. Math. Soc. 104 (1962), 62–78. MR 138128
- M. B. Marcus, Upper bounds for the asymptotic maxima of continuous Gaussian processes, Ann. Math. Statist. 43 (1972), 522–533. MR 388519
- Y. Mishura, G. Shevchenko, and S. Shklyar, Gaussian processes with Volterra kernels, Stochastic Processes, Statistical Methods, and Engineering Mathematics. SPAS 2019 (S. Silvestrov, A. Malyarenko, Y. Ni, and M. Rančić, eds.), Springer, Cham, 2022, pp. 249–276.
- Y. Mishura and S. Shklyar, Gaussian Volterra processes with power-type kernels. Part I, Mod. Stoch. Theory Appl. 9 (2022), no. 3, 313–338. MR 4462026
- —, Gaussian Volterra processes with power-type kernels. Part II, Mod. Stoch. Theory Appl. 9 (2022), no. 4, 431–452. MR 4510382
- I. Norros, E. Valkeila, and J. Virtamo, An elementary approach to a Girsanov formula and other analytical results on fractional Brownian motions, Bernoulli 5 (1999), no. 4, 571–587. MR 1704556
- K. V. Ralchenko, Approximation of multifractional Brownian motion by absolutely continuous processes, Theory Probab. Math. Statist. (2011), no. 82, 115–127. MR 2790487
- T. Sottinen and L. Viitasaari, Stochastic analysis of Gaussian processes via Fredholm representation, Int. J. Stoch. Anal. (2016), Art. ID 8694365, 15. MR 3536393
- A. Yazigi, Representation of self-similar Gaussian processes, Statist. Probab. Lett. 99 (2015), 94–100. MR 3321501
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Additional Information
Yuliya Mishura
Affiliation:
Department of Probability, Statistics and Actuarial Mathematics, Taras Shevchenko National University of Kyiv, 64/13, Volodymyrs’ka St., 01601 Kyiv, Ukraine
Email:
yuliyamishura@knu.ua
Kostiantyn Ralchenko
Affiliation:
Department of Probability, Statistics and Actuarial Mathematics, Taras Shevchenko National University of Kyiv, 64/13, Volodymyrs’ka St., 01601 Kyiv, Ukraine
Email:
kostiantynralchenko@knu.ua
Sergiy Shklyar
Affiliation:
Department of Probability, Statistics and Actuarial Mathematics, Taras Shevchenko National University of Kyiv, 64/13, Volodymyrs’ka St., 01601 Kyiv, Ukraine
Email:
sergiyshklyar@knu.ua
Keywords:
Gaussian Volterra process,
asymptotic growth,
long- and short-range dependence,
parameter estimation,
Ornstein–Uhlenbeck process
Received by editor(s):
September 24, 2022
Accepted for publication:
November 30, 2022
Published electronically:
May 2, 2023
Additional Notes:
The research by Yuliya Mishura has been supported by the Swedish Foundation for Strategic Research, grant no. UKR22–0017.
Kostiantyn Ralchenko is grateful to his hosts at Macquarie University, where he was a Visiting Fellow sponsored by Sydney Mathematical Research Institute (SMRI) under the Ukrainian Visitors Program.
Article copyright:
© Copyright 2023
Taras Shevchenko National University of Kyiv
