Asymptotic error distribution for the Euler scheme with locally Lipschitz coefficients
Asymptotic error distribution for the Euler scheme with locally Lipschitz coefficients
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DOI：
10.1016/j.spa.2019.07.003
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发表时间：
2017-09
影响因子：
1.4
通讯作者：
P. Protter;Lisha Qiu;Jaime San Martín
中科院分区：
文献类型：
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作者：
P. Protter;Lisha Qiu;Jaime San Martín
在求解随机微分方程数值格式的传统工作中，通常采用全局Lipschitz假设来保证不同类型的收敛性。在实践中，这通常是一个过于强烈的条件。在实际应用中，布朗运动驱动的SDE有时在紧集上的系数只有Lipschitz，但SDE解的路径可以是任意大的。本文证明了在约束较少的假设下，即局部Lipschitz和无有限时间爆炸情况下的一个弱收敛结果和在概率上的收敛性。证明了在全局Lipschitz条件下，如果一个数值格式在具有一定速率的紧时间集（UCP）上概率一致收敛，那么当将全局Lipschitz条件替换为局部Lipschitz +无有限爆炸条件时，具有相同速率的UCP成立。对于欧拉格式，还建立了误差过程的弱收敛性。本文的主要贡献是证明了归一化误差过程的n弱收敛性，并给出了极限过程。在全局Lipschitz和局部Lipschitz条件下，进一步研究了弱极限过程第二矩的有界性及其运行上极值。
In traditional works on numerical schemes for solving stochastic differential equations (SDEs), the globally Lipschitz assumption is often assumed to ensure different types of convergence. In practice, this is often too strong a condition. Brownian motion driven SDEs used in applications sometimes have coefficients which are only Lipschitz on compact sets, but the paths of the SDE solutions can be arbitrarily large. In this paper, we prove convergence in probability and a weak convergence result under a less restrictive assumption, that is, locally Lipschitz and with no finite time explosion. We prove if a numerical scheme converges in probability uniformly on any compact time set (UCP) with a certain rate under a global Lipschitz condition, then the UCP with the same rate holds when a globally Lipschitz condition is replaced with a locally Lipschitz plus no finite explosion condition. For the Euler scheme, weak convergence of the error process is also established. The main contribution of this paper is the proof of n weak convergence of the normalized error process and the limit process is also provided. We further study the boundedness of the second moments of the weak limit process and its running supremum under both global Lipschitz and locally Lipschitz conditions.