Stochastic representation and path properties of a fractional Cox–Ingersoll–Ross process
Authors:
Yu. S. Mishura, V. I. Piterbarg, K. V. Ralchenko and A. Yu. Yurchenko-Tytarenko
Translated by:
N. N. Semenov
Journal:
Theor. Probability and Math. Statist. 97 (2018), 167-182
MSC (2010):
Primary 60G22; Secondary 60G15, 60H10
DOI:
https://doi.org/10.1090/tpms/1055
Published electronically:
February 21, 2019
MathSciNet review:
3746006
Full-text PDF
Abstract |
References |
Similar Articles |
Additional Information
Abstract: We consider the Cox–Ingersoll–Ross process that satisfies the stochastic differential equation $dX_t = aX_t dt+\sigma \sqrt {X_t} dB^H_t$ driven by a fractional Brownian motion $B^H_t$ with the Hurst index exceeding $\frac {2}{3}$, where $\int _0^t\sqrt {X_s} dB^H_s$ is the pathwise integral defined as the limit of the corresponding Riemann–Stieltjes sums. We show that the Cox–Ingersoll–Ross process coincides with the square of the fractional Ornstein–Uhlenbeck process up to the first return to zero. Based on this observation, we consider the square of the fractional Ornstein–Uhlenbeck process with an arbitrary Hurst index and prove that it satisfies the above stochastic differential equation up to the first return to zero if $\int _0^t\sqrt {X_s} dB^H_s$ is understood as the pathwise Stratonovich integral. Then a natural question arises about the first visit to zero of the fractional Cox–Ingersoll–Ross process which coincides with the first visit to zero of the fractional Ornstein–Uhlenbeck process. Since the latter process is Gaussian, we use the bounds for the distributions of Gaussian processes to prove that the probability of a visit to zero over a finite time equals 1 if $a<0$. Otherwise this probability is positive. We provide an upper bound for this probability.
References
- V. Anh and A. Inoue, Financial markets with memory. I. Dynamic models, Stoch. Anal. Appl. 23 (2005), no. 2, 275–300. MR 2130350, DOI https://doi.org/10.1081/SAP-200050096
- Christian Bayer, Peter Friz, and Jim Gatheral, Pricing under rough volatility, Quant. Finance 16 (2016), no. 6, 887–904. MR 3494612, DOI https://doi.org/10.1080/14697688.2015.1099717
- Ji Xin Shi and Zhao Ben Fang, Long memory in the China stock market volatility: a bull and bear market perspective, J. Univ. Sci. Technol. China 42 (2012), no. 3, 179–184 (Chinese, with English and Chinese summaries). MR 2986340
- Patrick Cheridito, Hideyuki Kawaguchi, and Makoto Maejima, Fractional Ornstein-Uhlenbeck processes, Electron. J. Probab. 8 (2003), no. 3, 14. MR 1961165, DOI https://doi.org/10.1214/EJP.v8-125
- J. C. Cox, J. E. Ingersoll, and S. A. Ross, A re-examination of traditional hypotheses about the term structure of interest rates, J. Finance 36 (1981), 769–799.
- John C. Cox, Jonathan E. Ingersoll Jr., and Stephen A. Ross, An intertemporal general equilibrium model of asset prices, Econometrica 53 (1985), no. 2, 363–384. MR 785474, DOI https://doi.org/10.2307/1911241
- John C. Cox, Jonathan E. Ingersoll Jr., and Stephen A. Ross, A theory of the term structure of interest rates, Econometrica 53 (1985), no. 2, 385–407. MR 785475, DOI https://doi.org/10.2307/1911242
- Laurent Decreusefond and David Nualart, Hitting times for Gaussian processes, Ann. Probab. 36 (2008), no. 1, 319–330. MR 2370606, DOI https://doi.org/10.1214/009117907000000132
- Z. Ding, C. W. Granger, and R. F. Engle, A long memory property of stock market returns and a new model, J. Empirical Finance 1 (1993), no. 1, 83–106.
- Denis Feyel and Arnaud de la Pradelle, The FBM Ito’s formula through analytic continuation, Electron. J. Probab. 6 (2001), no. 26, 22. MR 1873303, DOI https://doi.org/10.1214/EJP.v6-99
- S. V. Kuchuk-Yatsenko, Yu. S. Mīshura, and Ē. Yu. Munchak, Application of Malliavin calculus to the exact and approximate estimation of options and assets under stochastic volatility, Teor. Ĭmovīr. Mat. Stat. 94 (2016), 93–115 (Ukrainian, with English, Russian and Ukrainian summaries); English transl., Theory Probab. Math. Statist. 94 (2017), 97–120. MR 3553457, DOI https://doi.org/10.1090/tpms/1012
- Sergii Kuchuk-Iatsenko and Yuliya Mishura, Pricing the European call option in the model with stochastic volatility driven by Ornstein-Uhlenbeck process. Exact formulas, Mod. Stoch. Theory Appl. 2 (2015), no. 3, 233–249. MR 3407504, DOI https://doi.org/10.15559/15-VMSTA36CNF
- Alexander Kukush, Yuliya Mishura, and Kostiantyn Ralchenko, Hypothesis testing of the drift parameter sign for fractional Ornstein-Uhlenbeck process, Electron. J. Stat. 11 (2017), no. 1, 385–400. MR 3608678, DOI https://doi.org/10.1214/17-EJS1237
- Pedro Lei and David Nualart, Stochastic calculus for Gaussian processes and application to hitting times, Commun. Stoch. Anal. 6 (2012), no. 3, 379–402. MR 2988698
- Nikolai N. Leonenko, Mark M. Meerschaert, and Alla Sikorskii, Correlation structure of fractional Pearson diffusions, Comput. Math. Appl. 66 (2013), no. 5, 737–745. MR 3089382, DOI https://doi.org/10.1016/j.camwa.2013.01.009
- Nikolai N. Leonenko, Mark M. Meerschaert, and Alla Sikorskii, Fractional Pearson diffusions, J. Math. Anal. Appl. 403 (2013), no. 2, 532–546. MR 3037487, DOI https://doi.org/10.1016/j.jmaa.2013.02.046
- Nicolas Marie, A generalized mean-reverting equation and applications, ESAIM Probab. Stat. 18 (2014), 799–828. MR 3334015, DOI https://doi.org/10.1051/ps/2014002
- Alexander Melnikov, Yuliya Mishura, and Georgiy Shevchenko, Stochastic viability and comparison theorems for mixed stochastic differential equations, Methodol. Comput. Appl. Probab. 17 (2015), no. 1, 169–188. MR 3306678, DOI https://doi.org/10.1007/s11009-013-9336-9
- Yuliya S. Mishura, Stochastic calculus for fractional Brownian motion and related processes, Lecture Notes in Mathematics, vol. 1929, Springer-Verlag, Berlin, 2008. MR 2378138
- G. M. Molchan, On the maximum of fractional Brownian motion, Teor. Veroyatnost. i Primenen. 44 (1999), no. 1, 111–115 (Russian, with Russian summary); English transl., Theory Probab. Appl. 44 (2000), no. 1, 97–102. MR 1751192, DOI https://doi.org/10.1137/S0040585X97977379
- Ivan Nourdin, Selected aspects of fractional Brownian motion, Bocconi & Springer Series, vol. 4, Springer, Milan; Bocconi University Press, Milan, 2012. MR 3076266
- Vladimir I. Piterbarg, Asymptotic methods in the theory of Gaussian processes and fields, Translations of Mathematical Monographs, vol. 148, American Mathematical Society, Providence, RI, 1996. Translated from the Russian by V. V. Piterbarg; Revised by the author. MR 1361884
- K. Yamasaki, L. Muchnik, S. Havlin, A. Bunde, and H. E. Stanley, Scaling and memory in volatility return intervals in financial markets, Proc. Nat. Acad. Sci. USA 102 (2005), no. 26, 9424–9428.
- Martina Zähle, On the link between fractional and stochastic calculus, Stochastic dynamics (Bremen, 1997) Springer, New York, 1999, pp. 305–325. MR 1678495, DOI https://doi.org/10.1007/0-387-22655-9_13
References
- V. Anh and A. Inoue, Financial markets with memory I: Dynamic models, Stoch. Anal. Appl. 23 (2005), no. 2, 275–300. MR 2130350
- C. Bayer, P. Friz, and J. Gatheral, Pricing under rough volatility, Quant. Finance 16 (2016), no. 6 887–904. MR 3494612
- T. Bollerslev and H. O. Mikkelsen, Modelling and pricing long memory in stock market volatility, J. Econometrics 73 (1996), no. 1, 151–184. MR 2986340
- P. Cheridito, H. Kawaguchi, and M. Maejima, Fractional Ornstein–Uhlenbeck processes, Electron. J. Probab 8 (2003), no. 3, 1–14. MR 1961165
- J. C. Cox, J. E. Ingersoll, and S. A. Ross, A re-examination of traditional hypotheses about the term structure of interest rates, J. Finance 36 (1981), 769–799.
- J. C. Cox, J. E. Ingersoll, and S. A. Ross, An intertemporal general equilibrium model of asset prices, Econometrica 53 (1985), no. 1, 363–384. MR 785474
- J. C. Cox, J. E. Ingersoll, and S. A. Ross, A theory of the term structure of interest rates, Econometrica 53 (1985), no. 2, 385–408. MR 785475
- L. Decreusefond and D. Nualart, Hitting times for Gaussian processes, Anal. Probab. 36 (2008), 319–330. MR 2370606
- Z. Ding, C. W. Granger, and R. F. Engle, A long memory property of stock market returns and a new model, J. Empirical Finance 1 (1993), no. 1, 83–106.
- D. Feyel and A. de la Pradelle, The FBM Itô’s formula through analytic continuation, Electron. J. Probab. 6 (2001), paper 26. MR 1873303
- S. Kuchuk-Iatsenko, Y. Mishura, and Y. Munchak, Application of Malliavin calculus to exact and approximate option pricing under stochastic volatility, Theory Probab. Math. Statist. 94 (2016), 93–115. MR 3553457
- S. Kuchuk-Iatsenko and Y. Mishura, Pricing the European call option in the model with stochastic volatility driven by Ornstein–Uhlenbeck process. Exact formulas, Mod. Stoch. Theory Appl. 2 (2015), no. 3, 233–249. MR 3407504
- A. Kukush, Y. Mishura, and K. Ralchenko, Hypothesis testing of the drift parameter sign for fractional Ornstein–Uhlenbeck process, Electron. J. Statist. 11 (2017), no. 1, 385–400. MR 3608678
- P. Lei and D. Nualart, Stochastic calculus for Gaussian processes and application to hitting times, Commun. Stoch. Anal. 6 (2012), no. 3, 379–402. MR 2988698
- N. Leonenko, M. Meerschaert, and A. Sikorskii, Correlation structure of fractional Pearson diffusion, Comput. Math. Appl. 66 (2013), no. 5, 737–745. MR 3089382
- N. Leonenko, M. Meerschaert, and A. Sikorskii, Fractional Pearson diffusion, J. Math. Anal. Appl. 403 (2013), no. 2, 532–546. MR 3037487
- N. Marie, A generalized mean-reverting equation and applications, ESAIM Probab. Stat. 18 (2014), 799–828. MR 3334015
- A. Melnikov, Y. Mishura, and G. Shevchenko, Stochastic viability and comparison theorems for mixed stochastic differential equations, Methodol. Comput. Appl. Probab. 17 (2015), no. 1, 169–188. MR 3306678
- Y. Mishura, Stochastic Calculus for Fractional Brownian Motion and Related Processes, Lecture Notes in Math., vol. 1929, Springer Science & Business Media, Berlin, 2008. MR 2378138
- G. M. Molchan, On the maximum of fractional Brownian motion, Theory Probab. Appl. 44 (2000), 97–102. MR 1751192
- I. Nourdin, Selected Aspects of Fractional Brownian Motion, Springer, New York, 2012. MR 3076266
- V. I. Piterbarg, Asymptotic Methods in Theory of Gaussian Random Processes and Fields, Transl. Math. Monogr., vol. 148, Amer. Math. Soc., Providence, 2012. MR 1361884
- K. Yamasaki, L. Muchnik, S. Havlin, A. Bunde, and H. E. Stanley, Scaling and memory in volatility return intervals in financial markets, Proc. Nat. Acad. Sci. USA 102 (2005), no. 26, 9424–9428.
- M. Zähle, On the link between fractional and stochastic calculus, Stochastic Dynamics (H. Grauel and M. Gundlach, eds.), Springer, New-York, 1999, pp. 305–325. MR 1678495
Similar Articles
Retrieve articles in Theory of Probability and Mathematical Statistics
with MSC (2010):
60G22,
60G15,
60H10
Retrieve articles in all journals
with MSC (2010):
60G22,
60G15,
60H10
Additional Information
Yu. S. Mishura
Affiliation:
Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, Taras Shevchenko National University of Kyiv, Volodymyrs’ka Street, 64/13, Kyiv 01601, Ukraine
Email:
myus@univ.kiev.ua
V. I. Piterbarg
Affiliation:
Laboratory of Probability Theory, Faculty for Mechanics and Mathematics, Moscow State University, Leninskie gory, 1, Moscow, 119991, Russian Federation
Email:
piter@mech.math.msu.su
K. V. Ralchenko
Affiliation:
Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, Taras Shevchenko National University of Kyiv, Volodymyrs’ka Street, 64/13, Kyiv 01601, Ukraine
Email:
k.ralchenko@gmail.com
A. Yu. Yurchenko-Tytarenko
Affiliation:
Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, Taras Shevchenko National University of Kyiv, Volodymyrs’ka Street, 64/13, Kyiv 01601, Ukraine
Email:
ayurty@gmail.com
Keywords:
Fractional Cox–Ingersoll–Ross process,
stochastic differential equation,
fractional Ornstein–Uhlenbeck process,
Stratonovich integral
Received by editor(s):
April 23, 2017
Published electronically:
February 21, 2019
Article copyright:
© Copyright 2019
American Mathematical Society