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 Published  August 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, 2016, 1 (0) : 4-. doi: 10.1186/s41546-016-0005-0
References:
[1]

Bismut, JM:Conjugate convex functions in optimal stochastic control. J. Math. Anal. Appl 44, 384-404(1973) Google Scholar

[2]

El Karoui, N, Peng, S, Quenez, M:Backward stochastic differential equations in finance. Math. Fin 7, 1-71 (1997) Google Scholar

[3]

Fisher, RA:Theory of statistical estimation. Proc. Cambridge Phylosophical Society 22, 700-725 (1925) Google Scholar

[4]

Freidlin, MI, Wentzell, AD Random Perturbations of Dynamical Systems, 2nd Ed. Springer, NY (1998) Google Scholar

[5]

Gasparyan, S, Kutoyants, YA:On approximation of the BSDE with unknown volatility in forward equation. Armenian J. Math 7(1), 59-79 (2015) Google Scholar

[6]

Ibragimov, IA, Has'minskii, RZ:Statistical Estimation-Asymptotic Theory. Springer, New York (1981) Google Scholar

[7]

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) Google Scholar

[8]

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) Google Scholar

[9]

Kutoyants, YA:Identification of Dynamical Systems with Small Noise. Kluwer Academic Publisher, Dordrecht (1994) Google Scholar

[10]

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) Google Scholar

[11]

Kutoyants, YA:On Multi-Step MLE-Process for Ergodic Diffusion. arXiv:1504.01869[math.ST] (2015) Google Scholar

[12]

Kutoyants, YA, Motrunich, A:On milti-step MLE-process for Markov sequences. Metrika 79(6), 705-724(2016) Google Scholar

[13]

Kutoyants, YA, Zhou, L:On approximation of the backward stochastic differential equation.(arXiv:1305.3728). J. Stat. Plann. Infer 150, 111-123 (2014) Google Scholar

[14]

Le Cam, L:On the asymptotic theory of estimation and testing hypotheses. Proc. 3rd Berkeley Symposium, vol. 1, pp. 129-156 (1956) Google Scholar

[15]

Lehmann, EL, Romano, JP Testing Statistical Hypotheses, 3rd ed. Springer, NY (2005) Google Scholar

[16]

Liptser, R, Shiryaev, AN Statistics of Random Processes, v.'s 1 and 2, 2-nd ed. Springer, NY (2001) Google Scholar

[17]

Pardoux, E, Peng, S:Adapted solution of a backward stochastic differential equation. System Control Letter 14, 55-61 (1990) Google Scholar

[18]

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) Google Scholar

[19]

Robinson, PM:The stochastic difference between econometric statistics. Econometrica 56(3), 531-548(1988) Google Scholar

[20]

Skorohod, AV, Khasminskii, RZ:On parameter estimation by indirect observations. Prob. Inform. Transm 32, 58-68 (1996) Google Scholar

[21]

Uchida, M, Yoshida, N:Adaptive Bayes type estimators of ergodic diffusion processes from discrete observations. Statist. Inference Stoch. Processes 17(2), 181-219 (2014) Google Scholar

show all references

References:
[1]

Bismut, JM:Conjugate convex functions in optimal stochastic control. J. Math. Anal. Appl 44, 384-404(1973) Google Scholar

[2]

El Karoui, N, Peng, S, Quenez, M:Backward stochastic differential equations in finance. Math. Fin 7, 1-71 (1997) Google Scholar

[3]

Fisher, RA:Theory of statistical estimation. Proc. Cambridge Phylosophical Society 22, 700-725 (1925) Google Scholar

[4]

Freidlin, MI, Wentzell, AD Random Perturbations of Dynamical Systems, 2nd Ed. Springer, NY (1998) Google Scholar

[5]

Gasparyan, S, Kutoyants, YA:On approximation of the BSDE with unknown volatility in forward equation. Armenian J. Math 7(1), 59-79 (2015) Google Scholar

[6]

Ibragimov, IA, Has'minskii, RZ:Statistical Estimation-Asymptotic Theory. Springer, New York (1981) Google Scholar

[7]

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) Google Scholar

[8]

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) Google Scholar

[9]

Kutoyants, YA:Identification of Dynamical Systems with Small Noise. Kluwer Academic Publisher, Dordrecht (1994) Google Scholar

[10]

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) Google Scholar

[11]

Kutoyants, YA:On Multi-Step MLE-Process for Ergodic Diffusion. arXiv:1504.01869[math.ST] (2015) Google Scholar

[12]

Kutoyants, YA, Motrunich, A:On milti-step MLE-process for Markov sequences. Metrika 79(6), 705-724(2016) Google Scholar

[13]

Kutoyants, YA, Zhou, L:On approximation of the backward stochastic differential equation.(arXiv:1305.3728). J. Stat. Plann. Infer 150, 111-123 (2014) Google Scholar

[14]

Le Cam, L:On the asymptotic theory of estimation and testing hypotheses. Proc. 3rd Berkeley Symposium, vol. 1, pp. 129-156 (1956) Google Scholar

[15]

Lehmann, EL, Romano, JP Testing Statistical Hypotheses, 3rd ed. Springer, NY (2005) Google Scholar

[16]

Liptser, R, Shiryaev, AN Statistics of Random Processes, v.'s 1 and 2, 2-nd ed. Springer, NY (2001) Google Scholar

[17]

Pardoux, E, Peng, S:Adapted solution of a backward stochastic differential equation. System Control Letter 14, 55-61 (1990) Google Scholar

[18]

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) Google Scholar

[19]

Robinson, PM:The stochastic difference between econometric statistics. Econometrica 56(3), 531-548(1988) Google Scholar

[20]

Skorohod, AV, Khasminskii, RZ:On parameter estimation by indirect observations. Prob. Inform. Transm 32, 58-68 (1996) Google Scholar

[21]

Uchida, M, Yoshida, N:Adaptive Bayes type estimators of ergodic diffusion processes from discrete observations. Statist. Inference Stoch. Processes 17(2), 181-219 (2014) Google Scholar

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