随机微分方程的自由纽结线性插值和Milstein格式
Free knot linear interpolation and the Milstein scheme for stochastic differential equations
摘要：
本文的主要目的是解决一个长期未解决的问题，关于随机微分方程的路径强逼近的全局误差在$L_{\infty}$ -范数。典型地，对于带有$k$离散点的标准逼近的SDEs，如分段内插的Ito-Taylor格式，具有平均超范数误差的阶数为$\left(\ln k/k\right)^{1/2}$。另一方面，有一个下界，这表明顺序$1/\sqrt{k}$是最好的可能样条逼近的SDES与$k$自由结。本文讨论如何得到一个可实现的方法，实现顺序$1/\sqrt{k}$的问题。到目前为止，关于这个问题的文献只给出了纯存在性结果，因此不适合于实际应用。本文介绍了一种非线性方法来逼近标量函数。该方法将Milstein格式与带自由结点的布朗运动的分段线性插值相结合，易于实现。此外，我们建立了尖锐的上下误差界与指定的常数表现出的影响的方程的系数。
英文摘要：
The main purpose of this paper is to give a solution to a long-standing unsolved problem concerning the pathwise strong approximation of stochastic differential equations with respect to the global error in the $L_{\infty}$-norm. Typically, one has average supnorm error of order $\left(\ln k/k\right)^{1/2}$ for standard approximations of SDEs with $k$ discretization points, like piecewise interpolated Ito-Taylor schemes. On the other hand there is a lower bound, which indicates that the order $1/\sqrt{k}$ is best possible for spline approximation of SDEs with $k$ free knots. The present paper deals with the question of how to get an implementable method, which achieves the order $1/\sqrt{k}$. Up to now, papers with regard to this issue give only pure existence results and so are inappropriate for practical use. In this paper we introduce a nonlinear method for approximating a scalar SDE. The method combines a Milstein scheme with a pieceweise linear interpolation of the Brownian motion with free knots and is easy to implement. Moreover, we establish sharp lower and upper error bounds with specified constants which exhibit the influence of the coefficients of the equation.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
