An adaptive Euler-Maruyama scheme for SDEs: convergence and stability
An adaptive Euler-Maruyama scheme for SDEs: convergence and stability
复制标题
SDE 的自适应 Euler-Maruyama 方案：收敛性和稳定性
DOI：
10.1093/imanum/drl032
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发表时间：
2006
影响因子：
2.1
通讯作者：
A. Stuart
中科院分区：
文献类型：
--
作者：
H. Lamba;Jonathan C. Mattingly;A. Stuart
对随机微分方程（SDE）的适应性算法的理解是一个开放区域，在该区域中，与算法的收敛性和稳定性（长期行为）相关的许多问题都无法解决。本文仅基于仅控制时间步的漂移组件，考虑了一种非常简单的自适应算法。研究了收敛和稳定性。收敛分析中的主要问题是，自适应方法不一定会将时间步骤驱动到用户输入公差零。必须量化这种可能性，并证明概率较低。稳定性分析的主要问题是牙术。假定噪声是非排定的，因此扩散过程是椭圆形的，并且假定漂移可以满足胁迫条件。然后，SDE在几何上是千古的（平均值迅速汇总到统计均衡）。如果漂移未线性界限，则显式固定时间步长近似，例如Euler-Maruyama方案，可能不会是ergodic。在这项工作中，可以证明简单的自适应时间步变策略可以解决此问题。除了证明恐怖性外，还证明了指数力矩的界限，从而推广了已知的SDE本身的结果。
The understanding of adaptive algorithms for stochastic differential equations (SDEs) is an open area, where many issues related to both convergence and stability (long-time behaviour) of algorithms are unresolved. This paper considers a very simple adaptive algorithm, based on controlling only the drift component of a time step. Both convergence and stability are studied. The primary issue in the convergence analysis is that the adaptive method does not necessarily drive the time steps to zero with the user-input tolerance. This possibility must be quantified and shown to have low probability. The primary issue in the stability analysis is ergodicity. It is assumed that the noise is nondegenerate, so that the diffusion process is elliptic, and the drift is assumed to satisfy a coercivity condition. The SDE is then geometrically ergodic (averages converge to statistical equilibrium exponentially quickly). If the drift is not linearly bounded, then explicit fixed time step approximations, such as the Euler-Maruyama scheme, may fail to be ergodic. In this work, it is shown that the simple adaptive time-stepping strategy cures this problem. In addition to proving ergodicity, an exponential moment bound is also proved, generalizing a result known to hold for the SDE itself.