Advertisement

Proceedings of the Steklov Institute of Mathematics

, Volume 287, Issue 1, pp 133–154 | Cite as

Approximation of the solution of the backward stochastic differential equation. Small noise, large sample and high frequency cases

  • Yury A. KutoyantsEmail author
Article

Abstract

We present a review of some recently obtained results on estimation of the solution of a backward stochastic differential equation (BSDE) in the Markovian case. We suppose that the forward equation depends on some finite-dimensional unknown parameter. We consider the problem of estimating this parameter and then use the proposed estimator to estimate the solution of the BSDE. This last estimator is constructed with the help of the solution of the corresponding partial differential equation. We are interested in three observation models admitting a consistent estimation of the unknown parameter: small noise, large samples and unknown volatility. In the first two cases we have a continuous time observation, and the unknown parameter is in the drift coefficient. In the third case the volatility of the forward equation depends on the unknown parameter, and we have discrete time observations. The presented estimators of the solution of the BSDE in the three casesmentioned are asymptotically efficient.

Keywords

STEKLOV Institute Maximum Likelihood Estimator Regularity Condition Fisher Information Matrix Small Noise 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    J.-M. Bismut, “Conjugate convex functions in optimal stochastic control,” J. Math. Anal. Appl. 44, 384–404 (1973).CrossRefMathSciNetGoogle Scholar
  2. 2.
    G. Dohnal, “On estimating the diffusion coefficient,” J. Appl. Probab. 24(1), 105–114 (1987).CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    M. I. Freidlin and A. D. Wentzell, Random Perturbations of Dynamical Systems, 2nd ed. (Springer, New York, 1998).CrossRefzbMATHGoogle Scholar
  4. 4.
    V. Genon-Catalot and J. Jacod, “On the estimation of the diffusion coefficient for multi-dimensional diffusion processes,” Ann. Inst. Henri Poincaré, Probab. Stat. 29(1), 119–151 (1993).zbMATHMathSciNetGoogle Scholar
  5. 5.
    N. El Karoui, S. Peng, and M. C. Quenez, “Backward stochastic differential equations in finance,” Math. Finance 7(1), 1–71 (1997).CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    K. Kamatani and M. Uchida, “Hybrid multi-step estimators for stochastic differential equations based on sampled data,” Stat. Inference Stoch. Processes, doi: 10.1007/s11203-014-9107-4 (2014).Google Scholar
  7. 7.
    Yu. Kutoyants, Identification of Dynamical Systems with Small Noise (Kluwer, Dordrecht, 1994).CrossRefzbMATHGoogle Scholar
  8. 8.
    Yu. A. Kutoyants, Statistical Inference for Ergodic Diffusion Processes (Springer, London, 2004).CrossRefzbMATHGoogle Scholar
  9. 9.
    Yu. A. Kutoyants and L. Zhou, “On approximation of the backward stochastic differential equation,” J. Stat. Plann. Inference 150, 111–123 (2014); arXiv: 1305.3728 [math.ST].CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    D. Levanony, A. Shwartz, and O. Zeitouni, “Recursive identification in continuous-time stochastic processes,” Stoch. Processes Appl. 49(2), 245–275 (1994).CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    R. S. Liptser and A. N. Shiryaev, Statistics of Random Processes, 2nd ed. (Springer, Berlin, 2001), Vol. 2.CrossRefGoogle Scholar
  12. 12.
    J. Ma and J. Yong, Forward-Backward Stochastic Differential Equations and Their Applications (Springer, Berlin, 1999), Lect. Notes Math. 1702.zbMATHGoogle Scholar
  13. 13.
    E. Pardoux and S. G. Peng, “Adapted solution of a backward stochastic differential equation,” Syst. Control Lett. 14(1), 55–61 (1990).CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    E. Pardoux and S. Peng, “Backward stochastic differential equations and quasilinear parabolic differential equations,” in Stochastic Partial Differential Equations and Their Applications (Springer, Berlin, 1992), Lect. Notes Control Inf. Sci. 176, pp. 200–217.CrossRefGoogle Scholar
  15. 15.
    M. Sørensen, “Estimating functions for diffusion-type processes,” in Statistical Methods for Stochastic Differential Equations, Ed. by M. Kessler, A. Lindner, and M. Sørensen (CRC Press, Boca Raton, FL, 2009), pp. 1–107.Google Scholar
  16. 16.
    L. Zhou, “Problèmes statistiques pour des EDS et les EDS rétrogrades,” PhD Thesis (Univ. Maine, Le Mans, 2013).Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2014

Authors and Affiliations

  1. 1.Laboratoire Manceau de MathématiquesUniversité du Maine, Le MansLe Mans Cedex 9France
  2. 2.International Laboratory of Quantitative FinanceNational Research University Higher School of EconomicsMoscowRussia

Personalised recommendations