Tail Distribution Estimates of Fractional CIR Model
Main Article Content
Abstract
The aim of this work is to study the tail distribution of the Cox–Ingersoll–Ross (CIR) model driven by fractional Brownian motion. We first prove the existence and uniqueness of the solution. Then based on the techniques of Malliavin calculus and a result established recently in [1], we obtain an explicit estimate for tail distributions.
Keywords:
CIR model, Fractional Brownian motion, Malliavin calculus.
References
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Vol. 128, No. 12, 2018, pp. 4154-4170, https://doi.org/10.1016/j.spa.2018.02.003.
V. Anh, A. Inoue, Financial Markets with Memory I: Dynamic models, J. Stoch. Anal. Appl, Vol. 23, No. 2, 2005,
pp. 275-300, https://doi.org/10.1081/SAP-200050096.
B. B. Mandelbrot, J. W. Van Ness, Fractional Brownian Motions, Fractional Noises and Applications, SIAM Rev., Vol. 10, 1968, pp. 422-437, https://doi.org/10.1137/1010093.
N. T. Dung, Fractional Stochastic Differential Equations with Applications to Finance, J. Math. Anal. Appl.,
Vol. 397, No. 1, 2013, pp. 334-348, https://doi.org/10.1016/j.jmaa.2012.07.062.
D. Nualart, The Malliavin Calculus and Related Topics, Probability and Its Applications, Springer-Verlag, Berlin, Second Edition, 2006, pp. 277-279
M. Za ̈hle, Integration with Respect to Fractal Functions and Stochastic Calculus, Part I, Probab, Theory Related Fields, Vol. 111, 1998, pp. 333-37, https://doi.org/10.1007/s004400050171.
Y. Hu, D. Nualart, X. Song, A Singular Stochastic Differential Equation Driven by Fractional Brownian Motion, Statist. Probab. Lett., Vol. 78, No. 14, 2008, pp. 2075-2085, https://doi.org/10.1016/j.spl.2008.01.080.
Y. Mishura, A. Y. Tytarenko, Fractional Cox–Ingersoll–Ross Process with Non-zero Mean, Theory and Applications, Vol. 5, No. 1, 2018, pp. 99-111, https://doi.org/10.15559/18-VMSTA97.
N. T. Dung, T. C. Son, Tail Distribution Estimates for One-dimensional Diffusion Processes, J. Math. Anal. Appl., Vol. 479, No. 2, 2019, pp. 2119-2138, https://doi.org/10.1016/j.spa.2018.02.003.
N. T. Dung, N. T. Hang, P. T. P. Thuy, Density Estimates for The Exponential Functionals of Fractional Brownian Motion, Comptes Rendus Mathematique, 202, https://doi.org/10.5802/crmath.274.
N. T. Dung, T. C. Son, T. M. Cuong, N. V. Tan, T. N. Quynh, Density Estimates for Solutions of Stochastic Functional Differential Equations, Acta Mathematica Scientia, Vol. 39B, No. 4, 2019, pp. 955-970, https://doi.org/10.1007/s10473-019-0404-2.
S. D. Marco, Smoothness and Asymptotic Estimates of Densities for SDEs with Locally Smooth Coefficients and Applications to Square Root-type Diffusions, Ann. Appl. Probab., Vol. 21, No. 4, 2011, pp. 1282-1321, https://doi.org/10.1214/10-AAP717.
N.T. Dung, Jacobi Processes Driven by Fractional Brownian Motion, Taiwanese J. Math., Vol. 18, No. 3, 2014, pp. 835-848, https://doi.org/10.11650/tjm.18.2014.3288.