Time-changed fractional Ornstein-Uhlenbeck process

References

[1] V. Anh, A. Inoue, Financial markets with memory I: Dynamic models. Stoch. Anal. Appl. 23, No 2 (2005), 275–300; 10.1081/SAP-200050096.Search in Google Scholar

[2] G. Ascione, Y. Mishura, E. Pirozzi, A fractional Ornstein-Uhlenbeck process with a stochastic forcing term and its applications. Methodol. Comput. Appl. (2019), 1–32; 10.1007/s11009-019-09748-y.Search in Google Scholar

[3] G. Ascione, E. Pirozzi, On a stochastic neuronal model integrating correlated inputs. Math. Biosci. Eng. 16, No 5 (2019), 5206–5225; 10.3934/mbe.2019260.Search in Google Scholar PubMed

[4] G. Ascione, E. Pirozzi, B. Toaldo, On the exit time from open sets of some semi-Markov processes. To appear in: Ann. Appl. Probab. (2019); arXiv:1709.06333.Search in Google Scholar

[5] G. Ascione, B. Toaldo, A semi-Markov leaky integrate-and-fire model. Mathematics7 (2019); 10.3390/math7111022.Search in Google Scholar

[6] J. Bertoin, Lévy Processes. Cambridge University Press, Cambridge (1996).Search in Google Scholar

[7] J. Bertoin, Subordinators: Examples and Applications. Springer, Berlin, (1999).Search in Google Scholar

[8] A. Brouste, S.M. Iacus, Parameter estimation from the discretely observed fractional Ornstein-Uhlenbeck process and the Yuima R package. Computation. Stat. 28, No 4 (2013), 1529–1547; 10.1007/s00180-012-0365-6.Search in Google Scholar

[9] Y.A. Butko, Chernoff approximation for semigroups generated by killed Feller processes and Feynman formulae for time-fractional Fokker–Planck–Kolmogorov equations. Fract. Calc. Appl. Anal. 21, No 5 (2018), 1203–1237; 10.1515/fca-2018-0065; https://www.degruyter.com/view/j/fca.2018.21.issue-5/issue-files/fca.2018.21.issue-5.xml.Search in Google Scholar

[10] D.O. Cahoy, F. Polito, Parameter estimation for fractional birth and fractional death processes. Stat. Comput. 24, No 2 (2014), 211–222; 10.1007/s11222-012-9365-1.Search in Google Scholar

[11] D.O. Cahoy, F. Polito, V. Phoha, Transient behavior of fractional queues and related processes. Methodol. Comput. Appl. 17, No 4 (2015), 739–759; 10.1007/s11009-013-9391-2.Search in Google Scholar

[12] P. Cheridito, H. Kawaguchi, M. Maejima, Fractional Ornstein-Uhlenbeck processes. Electron. J. Probab. 8(2003), 3–14; 10.1214/EJP.v8-125.Search in Google Scholar

[13] E. Cinlar, Markov additive processes and semi-regeneration. Technical Report, North-Western University, Center of Mathematical Studies in Economics and Management Science (1974).Search in Google Scholar

[14] M. Delorme, K.J. Wiese, Maximum of a fractional Brownian motion: Analytic results from perturbation theory. Phys. Rev. Lett. 115, No 21 (2015); 10.1103/PhysRevLett.115.210601.Search in Google Scholar PubMed

[15] M. D’Ovidio, S. Vitali, V. Sposini, O. Sliusarenko, P. Paradisi, G. Castellani, G. Pagnini, Centre-of-mass like superposition of Ornstein–Uhlenbeck processes: A pathway to non-autonomous stochastic differential equations and to fractional diffusion. Fract. Calc. Appl. Anal. 21, No 5 (2018), 1420–1435; 10.1515/fca-2018-0074; https://www.degruyter.com/view/j/fca.2018.21.issue-5/issue-files/fca.2018.21.issue-5.xml.Search in Google Scholar

[16] J. Gajda, A. Wyłomańska, Fokker-Planck type equations associated with fractional Borniwn motion controlled by infinitely divisible processes. Physica A405 (2014), 104–113; 10.1016/j.physa.2014.03.16.Search in Google Scholar

[17] J. Gajda, A. Wyłomańska, Time-changed Ornstein-Uhlenbeck process. J. Phys. A–Math. Theor. 48, No 13 (2015); 10.1088/1751-8113/48/13/135004.Search in Google Scholar

[18] J. Gatheral, T. Jaisson, M. Rosenbaum, Volatility is rough Quant. Financ. 18, No 6 (2018), 933–949; 10.1080/14697688.2017.1393551.Search in Google Scholar

[19] H. Gu, J.R. Liang, Y.X. Zhang, Time-changed geometric fractional Brownian motion and option pricing with transaction costs. Physica A405, No 15 (2012), 3971–3977; 10.1016/j.physa.2012.03.020.Search in Google Scholar

[20] Z. Guo, H. Yuan, Pricing European option under the time-changed mixed Brownian-fractional Brownian model. Physica A406 (2014), 73–79; 10.1016/j.physa.2014.03.032.Search in Google Scholar

[21] M.G. Hahn, K. Kobayashi, J. Ryvkina, S. Umarov, On time-changed Gaussian processes and their associated Fokker-Planck Kolmogorov equations. Electron. Commun. Prob. 16 (2011), 150–164; 10.1214/ECP.v16-1620.Search in Google Scholar

[22] Y. Hu, D. Nualart, Parameter estimation from fractional Ornstein-Uhlenbeck processes. Stat. Probabil. Lett. 80, No 11-12 (2010), 1030–1038; 10.1016/j.spl.2010.02.018.Search in Google Scholar

[23] J. Janssen, R. Manca, Semi-Markov Risk Models for Finance, Insurance and Reliability. Springer Science & Business Media (2007).Search in Google Scholar

[24] J.H. Jeon, A. V. Chechkin, R. Metzler, First passage behavior of multi-dimensional fractional Brownian motion and application to reaction phenomena. In: First-Passage Phenomena and Their Applications, World Scientific (2014), 175–202.10.1142/9789814590297_0008Search in Google Scholar

[25] A. Kukush, Y. Mishura, K. Ralchenko, Hypothesis testing of the drift parameter sign for fractional Ornstein-Uhlenbeck process. Electron. J. Stat. 11, No 1 (2017), 385–400; 10.1214/17-EJS1237.Search in Google Scholar

[26] A. Kumar, J. Gajda, A. Wyłomańska, R. Poloczanski, Fractional Brownian motion delayed by tempered and inverse tempered stable subordinators. Methodol. Comput. Appl. 21, No 1 (2019), 185–202; 10.1007/s11009-018-9648-x.Search in Google Scholar

[27] C. Lefèvre, M. Simon, SIR epidemics with stage of infection. Adv. Appl. Probab. 48, No 3 (2016), 768–791; 10.1017/apr.2016.27.Search in Google Scholar

[28] N.N. Leonenko, M.M. Meerschaert, A. Sikorskii, Fractional Pearson diffusions. J. Math. Anal. Appl. 403, No 2 (2013), 532–546; 10.1016/j.jmaa.2013.02.046.Search in Google Scholar PubMed PubMed Central

[29] N.N. Leonenko, M.M. Meerschaert, A. Sikorskii, Correlation structure of fractional Pearson diffusions. Comput. Math. Appl. 66, No 5 (2013), 737–745; 10.1016/j.camwa.2013.01.009.Search in Google Scholar PubMed PubMed Central

[30] X. Li, K. Zheng, Y. Yang, A simulation platform of DDoS attach based on network processor. In: 2008 International Conference on Computational Intelligence and Security, IEEE (2008), 421–426.10.1109/CIS.2008.190Search in Google Scholar

[31] M.M. Meerschaert, E. Nane, P. Vellaisamy, Distributed-order fractional diffusions on bounded domains. J. Math. Anal. Appl. 379, No 1 (2011), 216–228; 10.1016/j.jmaa.2010.12.056.Search in Google Scholar

[32] M.M. Meerschaert, H.P. Scheffler, Triangular array limits for continuous time random walks. Stoc. Proc. Appl. 118, No 9 (2008), 1606–1633; 10.1016/j.spa.2007.10.005.Search in Google Scholar

[33] M.M. Meerschaert, A. Sikorskii, Stochastic Models for Fractional Calculus. Walter de Gruyter, Berlin (2011).10.1515/9783110258165Search in Google Scholar

[34] M.M. Meerschaert, P. Straka, Inverse stable subordinators. Math. Model. Nat. Pheno. 8, No 2 (2013), 1–16; 10.1051/mmnp/20138201.Search in Google Scholar PubMed PubMed Central

[35] M.M. Meerschaert, B. Toaldo, Relaxation patterns and semi-Markov dynamics. Stoc. Proc. Appl. 129, (2018); 10.1016/j.spa.2018.08.004.Search in Google Scholar

[36] J.B. Mijena, Correlation structure of time-changed fractional Brownian motion. ArXiv preprint (2014); arxiv:1408.4502.Search in Google Scholar

[37] Y. Mishura, Stochastic Calculus for Fractional Brownian Motion and Related Processes. Springer, Berlin (2008).10.1007/978-3-540-75873-0Search in Google Scholar

[38] Y. Mishura, V.I. Piterbarg, K. Ralchenko, A. Yurchenko-Tytarenko, Stochastic representation and pathwise properties of fractional Cox-Ingersoll-Ross process. Theory Probab. Math. Stat. 97 (2018), 157–170; 10.1090/tpms/1055.Search in Google Scholar

[39] Y. Mishura, A. Yurchenko-Tytarenko, Fractional Cox-Ingersoll-Ross process with non-zero mean. Mod. Stoch. Theory Appl. 5 (2018), 99–111; 10.15559/18-VMSTA97.Search in Google Scholar

[40] W. Rudin, Real and Complex Analysis. McGraw-Hill (2006).Search in Google Scholar

[41] Y. Sakai, S. Funahashi, S. Shinomoto, Temporally correlated inputs to Leaky Integrate-and-Fire models can reproduce spiking statistics of cortical neurons. Neural Networks12, No 7–8 (1999), 1181–1190; 10.1016/s0893-6080(99)00053-2.Search in Google Scholar

[42] R.L. Schilling, R. Song, Z. Vondracek, Bernstein Functions: Theory and Applications. Walter de Gruyter, Berlin (2012).10.1515/9783110269338Search in Google Scholar

[43] S. Shinomoto, Y. Sakai, S. Funahashi, The Ornstein-Uhlenbeck process does not reproduce spiking statistics of neurons in prefrontal cortex. Neural Comput. 11, No 4 (1999), 935–951; 10.1162/089976699300016511.Search in Google Scholar PubMed

[44] J.A. Tenreiro Machado, V. Kiryakova, The chronicles of fractional calculus. Fract. Calc. Appl. Anal. 20, No 2 (2017), 307–336; 10.1515/fca-2017-0017; https://www.degruyter.com/view/j/fca.2017.20.issue-2/issue-files/fca.2017.20.issue-2.xml.Search in Google Scholar

[45] B. Toaldo, Convolution-type derivatives, hitting-times of subordinators and time-changed C⁰-semigroups. Potential Anal. 42, No 1 (2015), 115–140; 10.1007/s11118-014-9426-5.Search in Google Scholar

[46] A. Weron, M. Magdziarz, Anomalous diffusion and semimartingales. Europhys. Lett. 86, No 6 (2009); 10.1209/0295-5075/86/60010.Search in Google Scholar

[47] A. Wolfgang, J.K.B. Charles, H. Matthias, N. Frank, Vector-Valued Laplace Transform and Cauchy Problems. Birkhuser, Basel (2002).Search in Google Scholar