Stochastic differential equations with generalized stochastic volatility and statistical estimators
Authors:
M. Bel Hadj Khlifa, Yu. Mishura, K. Ralchenko, G. Shevchenko and M. Zili
Journal:
Theor. Probability and Math. Statist. 96 (2018), 1-13
MSC (2010):
Primary 60H10, 62F10, 62F12
DOI:
https://doi.org/10.1090/tpms/1030
Published electronically:
October 5, 2018
MathSciNet review:
3666868
Full-text PDF
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Additional Information
Abstract: We study a stochastic differential equation, the diffusion coefficient of which is a function of some adapted stochastic process. The various conditions for the existence and uniqueness of weak and strong solutions are presented. The drift parameter estimation in this model is investigated, and the strong consistency of the least squares and maximum likelihood estimators is proved. As an example, the Ornstein–Uhlenbeck model with stochastic volatility is considered.
References
- Y. Aït-Sahalia and R. Kimmel, Maximum likelihood estimation of stochastic volatility models, J. Financ. Econ. 83 (2007), 413–452.
- S. Altay and U. Schmock, Lecture notes on the Yamada–Watanabe condition for the pathwise uniqueness of solutions of certain stochastic differential equations, http://fam.tuwien.ac.at/~schmock/notes/Yamada-Watanabe.pdf, 2013.
- Meriem Bel Hadj Khlifa, Yuliya Mishura, Kostiantyn Ralchenko, and Mounir Zili, Drift parameter estimation in stochastic differential equation with multiplicative stochastic volatility, Mod. Stoch. Theory Appl. 3 (2016), no. 4, 269–285. MR 3593112, DOI https://doi.org/10.15559/16-VMSTA66
- Alexander S. Cherny and Hans-Jürgen Engelbert, Singular stochastic differential equations, Lecture Notes in Mathematics, vol. 1858, Springer-Verlag, Berlin, 2005. MR 2112227
- Jean-Pierre Fouque, George Papanicolaou, and K. Ronnie Sircar, Derivatives in financial markets with stochastic volatility, Cambridge University Press, Cambridge, 2000. MR 1768877
- J.-P. Fouque, G. Papanicolaou, and K. R. Sircar, Mean-reverting stochastic volatility, Int. J. Theor. Appl. Finance 3 (2000), no. 1, 101–142.
- S. Heston, A closed-form solution of options with stochastic volatility with applications to bond and currency options, The Review of Financial Studies 6 (1993), no. 2, 327–343.
- J. Hull and A. White, The pricing of options on assets with stochastic volatilities, J. Finance 42 (1987), 281–300.
- Sergii Kuchuk-Iatsenko and Yuliya Mishura, Option pricing in the model with stochastic volatility driven by Ornstein-Uhlenbeck process. Simulation, Mod. Stoch. Theory Appl. 2 (2015), no. 4, 355–369. MR 3456143, DOI https://doi.org/10.15559/15-VMSTA43
- 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
- Robert S. Liptser and Albert N. Shiryaev, Statistics of random processes. II, Second, revised and expanded edition, Applications of Mathematics (New York), vol. 6, Springer-Verlag, Berlin, 2001. Applications; Translated from the 1974 Russian original by A. B. Aries; Stochastic Modelling and Applied Probability. MR 1800858
- R. Sh. Liptser and A. N. Shiryayev, Theory of martingales, Mathematics and its Applications (Soviet Series), vol. 49, Kluwer Academic Publishers Group, Dordrecht, 1989. Translated from the Russian by K. Dzjaparidze [Kacha Dzhaparidze]. MR 1022664
- Makiko Nisio, Stochastic control theory, 2nd ed., Probability Theory and Stochastic Modelling, vol. 72, Springer, Tokyo, 2015. Dynamic programming principle. MR 3290231
- A. V. Skorokhod, Studies in the theory of random processes, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1965. Translated from the Russian by Scripta Technica, Inc. MR 0185620
- E. M. Stein and J. C. Stein, Stock price distributions with stochastic volatility: an analytic approach, Review of Financial Studies 4 (1991), no. 4, 727–752.
- Daniel W. Stroock and S. R. S. Varadhan, Diffusion processes with continuous coefficients. I, Comm. Pure Appl. Math. 22 (1969), 345–400. MR 253426, DOI https://doi.org/10.1002/cpa.3160220304
- Daniel W. Stroock and S. R. S. Varadhan, Diffusion processes with continuous coefficients. II, Comm. Pure Appl. Math. 22 (1969), 479–530. MR 254923, DOI https://doi.org/10.1002/cpa.3160220404
- Toshio Yamada and Shinzo Watanabe, On the uniqueness of solutions of stochastic differential equations, J. Math. Kyoto Univ. 11 (1971), 155–167. MR 278420, DOI https://doi.org/10.1215/kjm/1250523691
References
- Y. Aït-Sahalia and R. Kimmel, Maximum likelihood estimation of stochastic volatility models, J. Financ. Econ. 83 (2007), 413–452.
- S. Altay and U. Schmock, Lecture notes on the Yamada–Watanabe condition for the pathwise uniqueness of solutions of certain stochastic differential equations, http://fam.tuwien.ac.at/~schmock/notes/Yamada-Watanabe.pdf, 2013.
- M. Bel Hadj Khlifa, Y. Mishura, K. Ralchenko, and M. Zili, Drift parameter estimation in stochastic differential equation with multiplicative stochastic volatility, Mod. Stoch. Theory Appl. 3 (2016), no. 4, 269–285. MR 3593112
- A. S. Cherny and H.-J. Engelbert, Singular Stochastic Differential Equations, Lecture Notes in Mathematics, vol. 1858, Springer-Verlag, Berlin, 2005. MR 2112227
- J.-P. Fouque, G. Papanicolaou, and K. R. Sircar, Derivatives in Financial Markets with Stochastic Volatility, Cambridge University Press, Cambridge, 2000. MR 1768877
- J.-P. Fouque, G. Papanicolaou, and K. R. Sircar, Mean-reverting stochastic volatility, Int. J. Theor. Appl. Finance 3 (2000), no. 1, 101–142.
- S. Heston, A closed-form solution of options with stochastic volatility with applications to bond and currency options, The Review of Financial Studies 6 (1993), no. 2, 327–343.
- J. Hull and A. White, The pricing of options on assets with stochastic volatilities, J. Finance 42 (1987), 281–300.
- S. Kuchuk-Iatsenko and Y. Mishura, Option pricing in the model with stochastic volatility driven by Ornstein–Uhlenbeck process. Simulation, Mod. Stoch. Theory Appl. 2 (2015), no. 4, 355–369. MR 3456143
- 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
- R. S. Liptser and A. N. Shiryaev, Statistics of Random Processes. I, Applications of Mathematics, vol. 5, Springer-Verlag, Berlin, 2001. MR 1800858
- R. S. Liptser and A. N. Shiryayev, Theory of Martingales, Mathematics and its Applications (Soviet Series), vol. 49, Kluwer Academic Publishers Group, Dordrecht, 1989. MR 1022664
- M. Nisio, Stochastic Control Theory. Dynamic Programming Principle, Second edition, Probability Theory and Stochastic Modelling, vol. 72, Springer, Tokyo, 2015. MR 3290231
- A. V. Skorokhod, Studies in the Theory of Random Processes, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1965. MR 0185620
- E. M. Stein and J. C. Stein, Stock price distributions with stochastic volatility: an analytic approach, Review of Financial Studies 4 (1991), no. 4, 727–752.
- D. W. Stroock and S. R. S. Varadhan, Diffusion processes with continuous coefficients. I, Comm. Pure Appl. Math. 22 (1969), 345–400. MR 0253426
- D. W. Stroock and S. R. S. Varadhan, Diffusion processes with continuous coefficients. II, Comm. Pure Appl. Math. 22 (1969), 479–530. MR 0254923
- T. Yamada and S. Watanabe, On the uniqueness of solutions of stochastic differential equations, J. Math. Kyoto Univ. 11 (1971), 155–167. MR 0278420
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Additional Information
M. Bel Hadj Khlifa
Affiliation:
Department of Mathematics, Faculty of Sciences of Monastir, University of Monastir, Avenue de l’Environnement, 5000, Monastir, Tunisia
Email:
meriem.bhk@outlook.fr
Yu. Mishura
Affiliation:
Department of Probability Theory, Statistics and Actuarial Mathematics, Faculty of Mechanics and Mathematics, Taras Shevchenko National University of Kyiv, Volodymyrska, 64/13, Kyiv, Ukraine, 01601
Email:
myus@univ.kiev.ua
K. Ralchenko
Affiliation:
Department of Probability Theory, Statistics and Actuarial Mathematics, Faculty of Mechanics and Mathematics, Taras Shevchenko National University of Kyiv, Volodymyrska, 64/13, Kyiv, Ukraine, 01601
Email:
k.ralchenko@gmail.com
G. Shevchenko
Affiliation:
Department of Probability Theory, Statistics and Actuarial Mathematics, Faculty of Mechanics and Mathematics, Taras Shevchenko National University of Kyiv, Volodymyrska, 64/13, Kyiv, Ukraine, 01601
Email:
zhora@univ.kiev.ua
M. Zili
Affiliation:
Department of Mathematics, Faculty of Sciences of Monastir, University of Monastir, Avenue de l’Environnement, 5000, Monastir, Tunisia
Email:
Mounir.Zili@fsm.rnu.tn
Keywords:
Stochastic differential equation,
weak and strong solutions,
stochastic volatility,
drift parameter estimation,
maximum likelihood estimator,
strong consistency
Received by editor(s):
January 25, 2017
Published electronically:
October 5, 2018
Article copyright:
© Copyright 2018
American Mathematical Society