一类倒向随机微分方程多步格式的最优收敛阶估计
Sharp Convergence Rates of a Class of Multistep Formulae for Backward Stochastic Differential Equations
摘要: 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。
Abstract: 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.
文章引用：李恒杰, 任浩, 曾港. 一类倒向随机微分方程多步格式的最优收敛阶估计[J]. 应用数学进展, 2022, 11(5): 2538-2547. https://doi.org/10.12677/AAM.2022.115269

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