    January  2016, 1: 4 doi: 10.1186/s41546-016-0005-0

## On approximation of BSDE and multi-step MLE-processes

 Laboratoire Manceau des Mathématiques, Université du Maine Le Mans, France and National Research University “MPEI”, Moscow, Russia

Received  January 02, 2016 Revised  June 15, 2016

Fund Project: This work was done with partial financial support of the RSF grant number 14-49-10079.

We consider the problem of approximation of the solution of the backward stochastic differential equations in Markovian case. We suppose that the forward equation depends on some unknown finite-dimensional parameter. This approximation is based on the solution of the partial differential equations and multi-step estimator-processes of the unknown parameter. As the model of observations of the forward equation we take a diffusion process with small volatility. First we establish a lower bound on the errors of all approximations and then we propose an approximation which is asymptotically efficient in the sense of this bound. The obtained results are illustrated on the example of the Black and Scholes model.
Citation: Yu A. Kutoyants. On approximation of BSDE and multi-step MLE-processes. Probability, Uncertainty and Quantitative Risk, doi: 10.1186/s41546-016-0005-0
##### References:
  Bismut, JM:Conjugate convex functions in optimal stochastic control. J. Math. Anal. Appl 44, 384-404(1973),  El Karoui, N, Peng, S, Quenez, M:Backward stochastic differential equations in finance. Math. Fin 7, 1-71 (1997),  Fisher, RA:Theory of statistical estimation. Proc. Cambridge Phylosophical Society 22, 700-725 (1925),  Freidlin, MI, Wentzell, AD Random Perturbations of Dynamical Systems, 2nd Ed. Springer, NY (1998),  Gasparyan, S, Kutoyants, YA:On approximation of the BSDE with unknown volatility in forward equation. Armenian J. Math 7(1), 59-79 (2015),  Ibragimov, IA, Has'minskii, RZ:Statistical Estimation-Asymptotic Theory. Springer, New York (1981),  Jeganathan, P:Some asymptotic properties of risk functions when the limit of the experiment is mixed normal. Sankhya:The Indian Journal of Statistics 45(Series A, Pt.1), 66-87 (1983),  Kamatani, K, Uchida, M:Hybrid multi-step estimators for stochastic differential equations based on sampled data. Statist. Inference Stoch. Processes 18(2), 177-204 (2015),  Kutoyants, YA:Identification of Dynamical Systems with Small Noise. Kluwer Academic Publisher, Dordrecht (1994),  Kutoyants, YA:On approximation of the backward stochastic differential equation. Small noise, large samples and high frequency cases. Proceed. Steklov Inst. Mathematics 287, 133-154 (2014),  Kutoyants, YA:On Multi-Step MLE-Process for Ergodic Diffusion. arXiv:1504.01869[math.ST] (2015),  Kutoyants, YA, Motrunich, A:On milti-step MLE-process for Markov sequences. Metrika 79(6), 705-724(2016),  Kutoyants, YA, Zhou, L:On approximation of the backward stochastic differential equation.(arXiv:1305.3728). J. Stat. Plann. Infer 150, 111-123 (2014),  Le Cam, L:On the asymptotic theory of estimation and testing hypotheses. Proc. 3rd Berkeley Symposium, vol. 1, pp. 129-156 (1956),  Lehmann, EL, Romano, JP Testing Statistical Hypotheses, 3rd ed. Springer, NY (2005),  Liptser, R, Shiryaev, AN Statistics of Random Processes, v.'s 1 and 2, 2-nd ed. Springer, NY (2001),  Pardoux, E, Peng, S:Adapted solution of a backward stochastic differential equation. System Control Letter 14, 55-61 (1990),  Pardoux, E, Peng, S:Backward stochastic differential equations and quasilinear parabolic partial differential equations. Stochastic Partial Differential Equations and their Applications, pp. 200-217. Springer, Berlin (1992). (Lect. Notes Control Inf. Sci. 176),  Robinson, PM:The stochastic difference between econometric statistics. Econometrica 56(3), 531-548(1988),  Skorohod, AV, Khasminskii, RZ:On parameter estimation by indirect observations. Prob. Inform. Transm 32, 58-68 (1996),  Uchida, M, Yoshida, N:Adaptive Bayes type estimators of ergodic diffusion processes from discrete observations. Statist. Inference Stoch. Processes 17(2), 181-219 (2014),

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##### References:
  Bismut, JM:Conjugate convex functions in optimal stochastic control. J. Math. Anal. Appl 44, 384-404(1973),  El Karoui, N, Peng, S, Quenez, M:Backward stochastic differential equations in finance. Math. Fin 7, 1-71 (1997),  Fisher, RA:Theory of statistical estimation. Proc. Cambridge Phylosophical Society 22, 700-725 (1925),  Freidlin, MI, Wentzell, AD Random Perturbations of Dynamical Systems, 2nd Ed. Springer, NY (1998),  Gasparyan, S, Kutoyants, YA:On approximation of the BSDE with unknown volatility in forward equation. Armenian J. Math 7(1), 59-79 (2015),  Ibragimov, IA, Has'minskii, RZ:Statistical Estimation-Asymptotic Theory. Springer, New York (1981),  Jeganathan, P:Some asymptotic properties of risk functions when the limit of the experiment is mixed normal. Sankhya:The Indian Journal of Statistics 45(Series A, Pt.1), 66-87 (1983),  Kamatani, K, Uchida, M:Hybrid multi-step estimators for stochastic differential equations based on sampled data. Statist. Inference Stoch. Processes 18(2), 177-204 (2015),  Kutoyants, YA:Identification of Dynamical Systems with Small Noise. Kluwer Academic Publisher, Dordrecht (1994),  Kutoyants, YA:On approximation of the backward stochastic differential equation. Small noise, large samples and high frequency cases. Proceed. Steklov Inst. Mathematics 287, 133-154 (2014),  Kutoyants, YA:On Multi-Step MLE-Process for Ergodic Diffusion. arXiv:1504.01869[math.ST] (2015),  Kutoyants, YA, Motrunich, A:On milti-step MLE-process for Markov sequences. Metrika 79(6), 705-724(2016),  Kutoyants, YA, Zhou, L:On approximation of the backward stochastic differential equation.(arXiv:1305.3728). J. Stat. Plann. Infer 150, 111-123 (2014),  Le Cam, L:On the asymptotic theory of estimation and testing hypotheses. Proc. 3rd Berkeley Symposium, vol. 1, pp. 129-156 (1956),  Lehmann, EL, Romano, JP Testing Statistical Hypotheses, 3rd ed. Springer, NY (2005),  Liptser, R, Shiryaev, AN Statistics of Random Processes, v.'s 1 and 2, 2-nd ed. Springer, NY (2001),  Pardoux, E, Peng, S:Adapted solution of a backward stochastic differential equation. System Control Letter 14, 55-61 (1990),  Pardoux, E, Peng, S:Backward stochastic differential equations and quasilinear parabolic partial differential equations. Stochastic Partial Differential Equations and their Applications, pp. 200-217. Springer, Berlin (1992). (Lect. Notes Control Inf. Sci. 176),  Robinson, PM:The stochastic difference between econometric statistics. Econometrica 56(3), 531-548(1988),  Skorohod, AV, Khasminskii, RZ:On parameter estimation by indirect observations. Prob. Inform. Transm 32, 58-68 (1996),  Uchida, M, Yoshida, N:Adaptive Bayes type estimators of ergodic diffusion processes from discrete observations. Statist. Inference Stoch. Processes 17(2), 181-219 (2014),
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