American Institute of Mathematical Sciences

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
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References:
 [1] Rodrigue Gnitchogna Batogna, Abdon Atangana. Generalised class of Time Fractional Black Scholes equation and numerical analysis. Discrete & Continuous Dynamical Systems - S, 2019, 12 (3) : 435-445. doi: 10.3934/dcdss.2019028 [2] Gianni Gilioli, Sara Pasquali, Fabrizio Ruggeri. Nonlinear functional response parameter estimation in a stochastic predator-prey model. Mathematical Biosciences & Engineering, 2012, 9 (1) : 75-96. doi: 10.3934/mbe.2012.9.75 [3] David Lipshutz. Exit time asymptotics for small noise stochastic delay differential equations. Discrete & Continuous Dynamical Systems, 2018, 38 (6) : 3099-3138. doi: 10.3934/dcds.2018135 [4] Yan Wang, Lei Wang, Yanxiang Zhao, Aimin Song, Yanping Ma. A stochastic model for microbial fermentation process under Gaussian white noise environment. Numerical Algebra, Control & Optimization, 2015, 5 (4) : 381-392. doi: 10.3934/naco.2015.5.381 [5] Quan Zhou, Yabing Sun. High order one-step methods for backward stochastic differential equations via Itô-Taylor expansion. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021233 [6] Marjan Uddin, Hazrat Ali. Space-time kernel based numerical method for generalized Black-Scholes equation. Discrete & Continuous Dynamical Systems - S, 2020, 13 (10) : 2905-2915. doi: 10.3934/dcdss.2020221 [7] Rehana Naz, Imran Naeem. Exact solutions of a Black-Scholes model with time-dependent parameters by utilizing potential symmetries. Discrete & Continuous Dynamical Systems - S, 2020, 13 (10) : 2841-2851. doi: 10.3934/dcdss.2020122 [8] Azmy S. Ackleh, Jeremy J. Thibodeaux. Parameter estimation in a structured erythropoiesis model. Mathematical Biosciences & Engineering, 2008, 5 (4) : 601-616. doi: 10.3934/mbe.2008.5.601 [9] Hong Seng Sim, Wah June Leong, Chuei Yee Chen, Siti Nur Iqmal Ibrahim. Multi-step spectral gradient methods with modified weak secant relation for large scale unconstrained optimization. Numerical Algebra, Control & Optimization, 2018, 8 (3) : 377-387. doi: 10.3934/naco.2018024 [10] Kazeem Olalekan Aremu, Chinedu Izuchukwu, Grace Nnenanya Ogwo, Oluwatosin Temitope Mewomo. Multi-step iterative algorithm for minimization and fixed point problems in p-uniformly convex metric spaces. Journal of Industrial & Management Optimization, 2021, 17 (4) : 2161-2180. doi: 10.3934/jimo.2020063 [11] Shaokuan Chen, Shanjian Tang. Semi-linear backward stochastic integral partial differential equations driven by a Brownian motion and a Poisson point process. Mathematical Control & Related Fields, 2015, 5 (3) : 401-434. doi: 10.3934/mcrf.2015.5.401 [12] Robert Azencott, Yutheeka Gadhyan. Accurate parameter estimation for coupled stochastic dynamics. Conference Publications, 2009, 2009 (Special) : 44-53. doi: 10.3934/proc.2009.2009.44 [13] Alex Capaldi, Samuel Behrend, Benjamin Berman, Jason Smith, Justin Wright, Alun L. Lloyd. Parameter estimation and uncertainty quantification for an epidemic model. Mathematical Biosciences & Engineering, 2012, 9 (3) : 553-576. doi: 10.3934/mbe.2012.9.553 [14] Boling Guo, Guoli Zhou. On the backward uniqueness of the stochastic primitive equations with additive noise. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3157-3174. doi: 10.3934/dcdsb.2018305 [15] Erik Ekström, Johan Tysk. A boundary point lemma for Black-Scholes type operators. Communications on Pure & Applied Analysis, 2006, 5 (3) : 505-514. doi: 10.3934/cpaa.2006.5.505 [16] Kais Hamza, Fima C. Klebaner. On nonexistence of non-constant volatility in the Black-Scholes formula. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 829-834. doi: 10.3934/dcdsb.2006.6.829 [17] Ferenc Hartung. Parameter estimation by quasilinearization in differential equations with state-dependent delays. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1611-1631. doi: 10.3934/dcdsb.2013.18.1611 [18] Dariusz Borkowski. Forward and backward filtering based on backward stochastic differential equations. Inverse Problems & Imaging, 2016, 10 (2) : 305-325. doi: 10.3934/ipi.2016002 [19] Jasmina Djordjević, Svetlana Janković. Reflected backward stochastic differential equations with perturbations. Discrete & Continuous Dynamical Systems, 2018, 38 (4) : 1833-1848. doi: 10.3934/dcds.2018075 [20] Jan A. Van Casteren. On backward stochastic differential equations in infinite dimensions. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 803-824. doi: 10.3934/dcdss.2013.6.803

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