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一类倒向随机微分方程多步格式的最优收敛阶估计
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Abstract:
自解的存在唯一性问题被解决后，倒向随机微分方程逐步被应用于众多研究领域，例如随机最优控制、偏微分方程、金融数学、风险度量、非线性期望等。本文借助等距节点插值型求积公式的误差余项研究一类倒向随机微分方程多步格式的收敛阶问题。通过在空间层采用Chebyshev网格并结合重心Lagrange插值，提高了算法的数值精度。数值实验结果表明新提出的收敛阶是最优的。
Since the well-posedness of its solution was established, the backward stochastic differential equa-tion has been applied in many research fields, such as stochastic optimal control, partial differential equations, financial mathematics, risk measurement, nonlinear expectation and so on. This paper studies the convergence rate of a class of multistep formulae for backward stochastic differential equations with the help of remainder of the quadrature rule over the uniform grid. Due to the ap-plication of barycentric Lagrange interpolation with Chebyshev grids, computational accuracy of the multistep formula is greatly improved. Numerical results indicate that the proposed convergence order is sharp.
| [1] | Bismut, J.M. (1973) Conjugate Convex Functions in Optimal Stochastic Control. Journal of Mathematical Analysis and Applications, 44, 384-404. https://doi.org/10.1016/0022-247X(73)90066-8 |
| [2] | Pardoux, E. and Peng, S. (1990) Adapted Solution of a Backward Stochastic Differential Equation. Systems and Control Letters, 14, 55-61. https://doi.org/10.1016/0167-6911(90)90082-6 |
| [3] | Ma, J., Protter, P. and Yong, J. (1994) Solving For-ward-Backward Stochastic Differential Equations Explicitly—A Four Step Scheme. Probability Theory and Related Fields, 98, 339-359. https://doi.org/10.1007/BF01192258 |
| [4] | Peng, S. (1991) Probabilistic Interpretation for Sys-tems of Quasilinear Parabolic Partial Differential Equations. Stochastics and Stochastic Reports, 37, 61-74. https://doi.org/10.1080/17442509108833727 |
| [5] | Peng, S. (1990) A General Stochastic Maximum Principle for Optimal Control Problems. SIAM Journal on Control and Optimization, 28, 966-979. https://doi.org/10.1137/0328054 |
| [6] | Chevance, D. (1997) Numerical Methods for Backward Stochastic Differen-tial Equations. In: Rogers, L. and Talay, D., Eds., Numerical Methods in Finance, Cambridge University Press, Cam-bridge, 232-244.
https://doi.org/10.1017/CBO9781139173056.013 |
| [7] | Bender, C. and Denk, R. (2007) A Forward Scheme for Backward SDEs. Stochastic Processes and Their Applications, 117, 1793-1812. https://doi.org/10.1016/j.spa.2007.03.005 |
| [8] | Peng, S. (1999) A Linear Approximation Algorithm Using BSDE. Pacific Economic Review, 4, 285-292.
https://doi.org/10.1111/1468-0106.00079 |
| [9] | Mémin, J., Peng, S. and Xu, M.Y. (2008) Convergence of Solutions of Discrete Reflected Backward SDEs and Simulations. Acta Mathematicae Applicatae Sinica, English Series, 24, 1-18. https://doi.org/10.1007/s10255-006-6005-6 |
| [10] | Zhang, J. (2004) A Numerical Scheme for BSDEs. The Annals of Applied Probability, 14, 459-488.
https://doi.org/10.1214/aoap/1075828058 |
| [11] | Cvitanic, J. and Zhang, J. (2005) The Steepest Descent Method for Forward-Backward SDEs. Electronic Journal of Probability, 10, 1468-1495. https://doi.org/10.1214/EJP.v10-295 |
| [12] | Zhao, W., Chen, L. and Peng, S. (2006) A New Kind of Accurate Nu-merical Method for Backward Stochastic Differential Equations. SIAM Journal on Scientific Computing, 28, 1563-1581. https://doi.org/10.1137/05063341X |
| [13] | Zhao, W., Wang, J. and Peng, S. (2009) Error Estimates of the θ-Scheme for Backward Stochastic Differential Equations. Discrete and Continuous Dynamical Systems-B, 12, 905-924. https://doi.org/10.3934/dcdsb.2009.12.905 |
| [14] | Zhao, W., Fu, Y. and Zhou, T. (2014) New Kinds of High-Order Multistep Schemes for Coupled Forward Backward Stochastic Differential Equations. SIAM Journal on Scientific Com-puting, 36, A1731-A1751.
https://doi.org/10.1137/130941274 |
| [15] | Chassagneux, J.F. (2014) Linear Multistep Schemes for BSDEs. SIAM Journal on Numerical Analysis, 52, 2815-2836.
https://doi.org/10.1137/120902951 |
| [16] | Chassagneux, J.F. and Crisan, D. (2014) Runge-Kutta Schemes for Back-ward Stochastic Differential Equations. The Annals of Applied Probability, 24, 679-720. https://doi.org/10.1214/13-AAP933 |
| [17] | Tang, T., Zhao, W. and Zhou, T. (2017) Deferred Correction Methods for Forward Backward Stochastic Differential Equations. Numerical Mathematics: Theory, Methods and Applications, 10, 222-242.
https://doi.org/10.4208/nmtma.2017.s02 |
| [18] | Zhang, C., Wu, J. and Zhao, W. (2019) One-Step Multi-Derivative Methods for Backward Stochastic Differential Equations. Numerical Mathematics: Theory, Methods and Applications, 12, 1213-1230.
https://doi.org/10.4208/nmtma.OA-2018-0122 |
| [19] | Zhou, Q. and Sun, Y. (2021) High Order One-step Methods for Backward Stochastic Differential Equations via It？-Taylor Expansion. Discrete and Continuous Dynamical Sys-tems-B. https://doi.org/10.3934/dcdsb.2021233 |
| [20] | Zhao, W., Zhang, G. and Ju, L. (2010) A Stable Multistep Scheme for Solving Backward Stochastic Differential Equations. SIAM Journal on Numerical Analysis, 48, 1369-1394. https://doi.org/10.1137/09076979X |
| [21] | Berrut, J.P. and Trefethen, L.N. (2004) Barycentric Lagrange Interpolation. SIAM Review, 46, 501-517.
https://doi.org/10.1137/S0036144502417715 |
