Gaussian Volterra processes with power-type kernels. Part II
Volume 9, Issue 4 (2022), pp. 431–452
Pub. online: 5 July 2022
Type: Research Article
Open Access
Open Access
Received
21 March 2022
21 March 2022
Revised
9 June 2022
9 June 2022
Accepted
17 June 2022
17 June 2022
Published
5 July 2022
5 July 2022
Abstract
In this paper the study of a three-parametric class of Gaussian Volterra processes is continued. This study was started in Part I of the present paper. The class under consideration is a generalization of a fractional Brownian motion that is in fact a one-parametric process depending on Hurst index H. On the one hand, the presence of three parameters gives us a freedom to operate with the processes and we get a wider application possibilities. On the other hand, it leads to the need to apply rather subtle methods, depending on the intervals where the parameters fall. Integration with respect to the processes under consideration is defined, and it is found for which parameters the processes are differentiable. Finally, the Volterra representation is inverted, that is, the representation of the underlying Wiener process via Gaussian Volterra process is found. Therefore, it is shown that for any indices for which Gaussian Volterra process is defined, it generates the same flow of sigma-fields as the underlying Wiener process – the property that has been used many times when considering a fractional Brownian motion.
References
Azmoodeh, E., Sottinen, T., Viitasaari, L., Yazigi, A.: Necessary and sufficient conditions for Hölder continuity of Gaussian processes. Statist. Probab. Letters 94, 230–235 (2014). MR3257384. https://doi.org/10.1016/j.spl.2014.07.030
Biagini, F., Hu, Y., Øksendal, B., Zhang, T.: Stochastic Calculus for Fractional Brownian Motion and Applications. Springer, London (2008). MR2387368. https://doi.org/10.1007/978-1-84628-797-8
Decreusefond, L.: Stochastic integration with respect to Volterra processes. Ann. I. H. Poincaré (B) Probab. Statist. 41(2), 123–149 (2005). MR2124078. https://doi.org/10.1016/j.anihpb.2004.03.004
Huang, S.T., Cambanis, S.: Stochastic and multiple Wiener integrals for Gaussian processes. Ann. Probab. 6(4), 585–614 (1978). MR0496408. https://doi.org/10.1214/aop/1176995480
Jost, C.: Transformation formulas for fractional Brownian motion. Stochastic Process. Appl. 116(10), 1341–1357 (2006). MR2260738. https://doi.org/10.1016/j.spa.2006.02.006
Mishura, Yu., Shevchenko, G.: Theory and Statistical Applications of Stochastic Processes. ISTE, London; Wiley, Hoboken (2017). https://doi.org/10.1002/9781119441601
Mishura, Yu., Shklyar, S.: Gaussian Volterra processes with power-type kernels. Part 1. Mod. Stoch. Theory Appl. (2022). https://doi.org/10.15559/22-VMSTA205
Mishura, Yu., Shevchenko, G., Shklyar, S.: Gaussian processes with Volterra kernels. In: Silvestrov, S., Malyarenko, A., Rancic, M. (eds.) Stochastic Processes, Stochastic Methods and Engineering Mathematics. Springer (2022) (to appear) arxiv:2001.03405.
Mishura, Yu.S.: Stochastic Calculus for Fractional Brownian Motion and Related Processes. Lecture Notes in Mathematics, vol. 1929. Springer, Berlin (2008). MR2378138. https://doi.org/10.1007/978-3-540-75873-0
Norros, I., Valkeila, E., Virtamo, J.: An elementary approach to a Girsanov formula and other analytical results on fractional Brownian motions. Bernoulli 5(4), 571–587 (1999). MR1704556. https://doi.org/10.2307/3318691
Nualart, D.: Stochastic calculus with respect to fractional Brownian motion. Annales de la Faculté des sciences de Toulouse: Mathématiques Ser. 6, 15(1), 63–78 (2006). MR2225747. https://doi.org/10.5802/afst.1113
Pipiras, V., Taqqu, M.S.: Are classes of deterministic integrands for fractional Brownian motion on an interval complete? Bernoulli 7(6), 873–897 (2001). MR1873833. https://doi.org/10.2307/3318624
Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives. Gordon and Breach, Reading (1993). MR1347689
Sottinen, T., Viitasaari, L.: Stochastic analysis of Gaussian processes via Fredholm representation. Int. J. Stoch. Anal. 2016, Article ID 8694365 (2016). MR3536393. https://doi.org/10.1155/2016/8694365
