2017
2016
2015
2014
2013
2012
2011
2010
2009
2008
2007
2006
2005
2004
2003
2002
2001
2000
1999
1998
1997
1996
1995
1994
1993
1992
1991
1990
1989
1988
1987
1986
1985
1984
1983
1982
1981
1980
1979
1978
1977
1976
1975
1974
1973
1972
1971
1970


Archive

About   Editors Board   Contacts   Template   Publication Ethics   Peer Review Process

Theory of Probability and Mathematical Statistics
(Teoriya imovirnostey ta matematychna statystyka)



STOCHASTIC DIFFERENTIAL EQUATIONS WITH GENERALIZED STOCHASTIC VOLATILITY AND STATISTICAL ESTIMATORS

M. BEL HADJ KHLIFA, YU. MISHURA, K. RALCHENKO, G. SHEVCHENKO, M. ZILI

Download PDF

Abstract: We study a stochastic di erential equation, the di usion coecient 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.

Keywords: Stochastic di erential equation, weak and strong solutions, stochastic volatility, drift parameter estimation, maximum likelihood estimator, strong consistency

Bibliography:
1. Y. At-Sahalia and R. Kimmel, Maximum likelihood estimation of stochastic volatility models, J. Financ. Econ. 83 (2007), 413-452.
2. S. Altay and U. Schmock, Lecture notes on the Yamada{Watanabe condition for the pathwise uniqueness of solutions of certain stochastic di erential equations, http://fam.tuwien.ac.at/~schmock/notes/Yamada-Watanabe.pdf, 2013.
3. M. Bel Hadj Khlifa, Y. Mishura, K. Ralchenko, and M. Zili, Drift parameter estimation in stochastic di erential equation with multiplicative stochastic volatility, Mod. Stoch. Theory Appl. 3 (2016), no. 4, 269-285.
4. A. S. Cherny and H.-J. Engelbert, Singular Stochastic Di erential Equations, Volume 1858 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2005.
5. J.-P. Fouque, G. Papanicolaou, and K. R. Sircar, Derivatives in Financial Markets with Stochastic Volatility, Cambridge University Press, Cambridge, 2000.
6. J.-P. Fouque, G. Papanicolaou, and K. R. Sircar, Mean-reverting stochastic volatility, Int. J. Theor. Appl. Finance 3 (2000), no. 1, 101-142.
7. 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.
8. J. Hull and A. White, The pricing of options on assets with stochastic volatilities, J. Finance 42 (1987), 281-300.
9. 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.
10. 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.
11. R. S. Liptser and A. N. Shiryaev, Statistics of Random Processes. I, Volume 5 of Applications of Mathematics, Springer-Verlag, Berlin, 2001.
12. R. S. Liptser and A. N. Shiryayev, Theory of Martingales, Volume 49 of Mathematics and its Applications (Soviet Series), Kluwer Academic Publishers Group, Dordrecht, 1989.
13. M. Nisio, Stochastic Control Theory. Dynamic Programming Principle, second edition, Volume 72 of Probability Theory and Stochastic Modelling, Springer, Tokyo, 2015.
14. A. V. Skorokhod. Studies in the Theory of Random Processes, Addison-Wesley Publishing Co.,Inc., Reading, Mass., 1965.
15. E. M. Stein and J. C. Stein, Stock price distributions with stochastic volatility: an analytic approach, Review of nancial Studies 4 (1991), no. 4, 727-752.
16. D. W. Stroock and S. R. S. Varadhan, Di usion processes with continuous coecients. I, Comm. Pure Appl. Math. 22 (1969), 345-400.
17. D. W. Stroock and S. R. S. Varadhan, Di usion processes with continuous coecients. II, Comm. Pure Appl. Math. 22 (1969), 479-530.
18. T. Yamada and S.Watanabe, On the uniqueness of solutions of stochastic di erential equations, J. Math. Kyoto Univ. 11 (1971), 155-167.