具有加性噪声和时间奇点的随机微分方程强解的有效变步长逼近

H. Hughes, Pathiranage Lochana Siriwardena
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引用次数: 1

摘要

我们考虑了具有加性噪声的随机微分方程,以及这些方程的系数允许漂移系数存在时间奇点的条件。给定最大步长,我们指定变量(自适应)步长,相对于它随着时间节点点接近奇点而减小。我们使用欧拉型数值格式来产生近似解并估计近似中的误差。当解被限制在一个不含奇异点的固定闭时间区间内时,得到了全局点向阶误差。当逼近到奇点范围内的一段时间内,为适当选择指数,可得到任意一阶误差。我们将此格式应用于布朗桥,它被定义为所考虑的一类随机微分方程的非预期解。在这种特殊情况下,我们证明了全局逐点误差是有序的,与考虑的近似有多接近奇点无关。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Efficient Variable Step Size Approximations for Strong Solutions of Stochastic Differential Equations with Additive Noise and Time Singularity
We consider stochastic differential equations with additive noise and conditions on the coefficients in those equations that allow a time singularity in the drift coefficient. Given a maximum step size, , we specify variable (adaptive) step sizes relative to which decrease as the time node points approach the singularity. We use an Euler-type numerical scheme to produce an approximate solution and estimate the error in the approximation. When the solution is restricted to a fixed closed time interval excluding the singularity, we obtain a global pointwise error of order . An order of error for any is obtained when the approximation is run up to a time within of the singularity for an appropriate choice of exponent . We apply this scheme to Brownian bridge, which is defined as the nonanticipating solution of a stochastic differential equation of the type under consideration. In this special case, we show that the global pointwise error is of order , independent of how close to the singularity the approximation is considered.
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