Mean-square convergence rates of stochastic theta methods for SDEs under a coupled monotonicity condition
Mean-square convergence rates of stochastic theta methods for SDEs under a coupled monotonicity condition
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耦合单调性条件下 SDE 随机 theta 方法的均方收敛率
DOI：
10.1007/s10543-019-00793-0
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发表时间：
2020
影响因子：
1.5
通讯作者：
Dong Bozhang
中科院分区：
文献类型：
--
作者：
Wang Xiaojie;Wu Jiayi;Dong Bozhang
本文回顾了求解具有非整体Lipschitz漂移和扩散系数的随机微分方程的著名随机theta方法(STM)。在区域内耦合单调的条件下，我们提出了一种新的方法，该方法只涉及精确解的求解过程，从而获得了具有该方法参数的STM的均方误差上界。这使我们能够很容易地恢复所考虑的格式的均方收敛速度，而不需要数值逼近的先验高阶矩估计。作为误差界的应用，在进一步的全局多项式增长条件下，我们得到了三种噪声驱动的随机微分方程的均方收敛速度。特别地，利用误差界分析了小噪声条件下SDE的逼近。结果表明，随机梯形算法比其他算法具有更好的收敛性能。此外，我们将STMS应用于取值于区域上的Ait-Sahalia型利率模型，并成功地证明了STM的收敛速度，即使在一般的临界情况下也是如此。这填补了Szpruch等人留下的空白。(bit Numer Math 51(2)：405-425,2011)，证明了向后欧拉法在非临界情况下的强收敛，但没有给出收敛速度。
The present article revisits the well-known stochastic theta methods (STMs) for stochastic differential equations (SDEs) with non-globally Lipschitz drift and diffusion coefficients. Under a coupled monotonicity condition in a domain, we propose a novel approach to achieve upper mean-square error bounds for STMs with the method parameters, which only get involved with the exact solution processes. This enables us to easily recover mean-square convergence rates of the considered schemes, without requiring a priori high-order moment estimates of numerical approximations. As applications of the error bounds, we derive mean-square convergence rates of STMs for SDEs driven by three kinds of noises under further globally polynomial growth condition. In particular, the error bounds are utilized to analyze approximation of SDEs with small noise. It is shown that the stochastic trapezoid formula gives better convergence performance than the other STMs. Furthermore, we apply STMs to the Ait-Sahalia-type interest rate model taking values in the domain, and successfully identify a convergence rate of order one-half for STMs with, even in a general critical case. This fills the gap left by Szpruch et al. (BIT Numer Math 51(2):405–425, 2011), where strong convergence of the backward Euler method was proved, without revealing a rate of convergence, for the model in a non-critical case.