## Abstract and Applied Analysis

### Parameter Estimation for Stochastic Differential Equations Driven by Mixed Fractional Brownian Motion

#### Abstract

We study the asymptotic properties of minimum distance estimator of drift parameter for a class of nonlinear scalar stochastic differential equations driven by mixed fractional Brownian motion. The consistency and limit distribution of this estimator are established as the diffusion coefficient tends to zero under some regularity conditions.

#### Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2014), Article ID 942307, 6 pages.

Dates
First available in Project Euclid: 6 October 2014

https://projecteuclid.org/euclid.aaa/1412607114

Digital Object Identifier
doi:10.1155/2014/942307

Mathematical Reviews number (MathSciNet)
MR3248882

Zentralblatt MATH identifier
07023363

#### Citation

Song, Na; Liu, Zaiming. Parameter Estimation for Stochastic Differential Equations Driven by Mixed Fractional Brownian Motion. Abstr. Appl. Anal. 2014, Special Issue (2014), Article ID 942307, 6 pages. doi:10.1155/2014/942307. https://projecteuclid.org/euclid.aaa/1412607114

#### References

• X. Mao, Stochastic Differential Equations and Applications, Elsevier, 2007.
• J. Bao and Z. Hou, “Existence of mild solutions to stochastic neutral partial functional differential equations with non-Lipschitz coefficients,” Computers & Mathematics with Applications, vol. 59, no. 1, pp. 207–214, 2010.
• J. Bao, Z. Hou, and C. Yuan, “Stability in distribution of mild solutions to stochastic partial differential equations,” Proceedings of the American Mathematical Society, vol. 138, no. 6, pp. 2169–2180, 2010.
• B. L. S. Prakasa Rao, Statistical Inference for Diffusion Type Processes, London, UK, Oxford University Press, 1999.
• Y. A. Kutoyants, Statistical Inference for Ergodic Diffusion Processes, Springer, London, UK, 2004.
• K. Bertin, S. Torres, and C. A. Tudor, “Drift parameter estimation in fractional diffusions driven by perturbed random walks,” Statistics & Probability Letters, vol. 81, no. 2, pp. 243–249, 2011.
• H. Yaozhong, N. David, X. Weilin, and Z. Weiguo, “Exact maximum likelihood estimator for drift fractional Brownian motion at discrete observation,” Acta Mathematica Scientia, vol. 31, no. 5, pp. 1851–1859, 2011.
• Y. Hu and D. Nualart, “Parameter estimation for fractional Ornstein-Uhlenbeck processes,” Statistics & Probability Letters, vol. 80, no. 11-12, pp. 1030–1038, 2010.
• B. L. S. P. Rao, Statistical Inference for Fractional Diffusion Processes, Wiley, 2011.
• C. Patrick, “Mixed fractional brownian motion,” Bernoulli, vol. 7, no. 6, pp. 913–934, 2001.
• A. W. Lo, “Maximum likelihood estimation of generalized Itô processes with discretely sampled data,” Econometric Theory, vol. 4, no. 2, pp. 231–247, 1988.
• W. C. Parr and W. R. Schucany, “Minimum distance estimation and components of goodness-of-fit statistics,” Journal of the Royal Statistical Society B: Methodological, vol. 44, no. 2, pp. 178–189, 1982.
• P. W. Millar, “A general approach to the optimality of minimum distance estimators,” Transactions of the American Mathematical Society, vol. 286, no. 1, pp. 377–418, 1984.
• D. L. Donoho and R. C. Liu, “The “automatic” robustness of minimum distance functionals,” The Annals of Statistics, vol. 16, no. 2, pp. 552–586, 1988.
• Y. A. Kutoyants, Identification of Dynamical Systems with Small Noise, Kluwer Academic, Dordrecht, The Netherlands, 1994.
• Y. Kutoyants and P. Pilibossian, “On minimum \emphL$_{1}$-norm estimate of the parameter of the Ornstein-Uhlenbeck process,” Statistics & Probability Letters, vol. 20, no. 2, pp. 117–123, 1994.
• S. Hénaff, “Asymptotics of a minimum distance estimator of the parameter of the Ornstein-Uhlenbeck process,” Comptes Rendus de l'Académie des Sciences I, vol. 325, no. 8, pp. 911–914, 1997.
• B. L. S. Prakasa Rao, “Minimum ${l}_{1}$-norm estimation for fractional Ornstein-Uhlenbeck type process,” Theory of Probability and Mathematical Statistics, vol. 71, pp. 160–168, 2004.
• L. Kouame, M. N'Zi, and A. F. Yode, “Asymptotics of minimum distance estimator of the parameter of stochastic process driven by a fractional Brownian motion,” Random Operators and Stochastic Equations, vol. 16, no. 4, pp. 399–407, 2008.
• M. Zili, “On the mixed fractional Brownian motion,” Journal of Applied Mathematics and Stochastic Analysis, vol. 2006, Article ID 32435, 9 pages, 2006.
• Y. Miao, “Minimum ${L}_{1}$-norm estimation for mixed fractional Ornstein-Uhlenbeck type process,” Acta Mathematica Vietnamica, vol. 35, no. 3, pp. 379–386, 2010.
• W.-L. Xiao, W. G. Z. Zhang, and X. L. Zhang, “Maximum-likelihood estimators in the mixed fractional Brownian motion,” Statistics, vol. 45, no. 1, pp. 73–85, 2011.
• F. Biagini, Y. Hu, B. ${\text{\O}}$ksendal, and T. Zhang, Stochastic Calculus for Fractional Brownian Motion and Applications, Springer, New York, NY, USA, 2008.
• A. A. Novikov, “The moment inequalities for stochastic integrals,” Theory of Probability Its Applications, vol. 16, no. 3, pp. 538–541, 1971.
• D. L. Burkholder, “Distribution function inequalities for martingales,” The Annals of Probability, vol. 16, no. 3, pp. 19–42, 1973.
• A. A. Novikov and E. Valkeila, “On some maximal inequalities for fractional brownian motions,” Statistics & Probability Letters, vol. 44, no. 1, pp. 47–54, 1999.
• Y. A. Kutoyants, Parameter Estimation for Stochastic Processes, Heldermann, Berlin, Germany, 1984. \endinput