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 $$D \subset {{\mathbb {R}}}^d, d \in {{\mathbb {N}}}$$ D ⊂ R d , d ∈ N , we propose a novel approach to achieve upper mean-square error bounds for STMs with the method parameters $$\theta \in [\tfrac{1}{2}, 1]$$ θ ∈ [ 1 2 , 1 ] , 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 $$D = ( 0, \infty )$$ D = ( 0 , ∞ ) , and successfully identify a convergence rate of order one-half for STMs with $$\theta \in [\tfrac{1}{2}, 1]$$ θ ∈ [ 1 2 , 1 ] , 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.

中文翻译：

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耦合单调性条件下 SDE 的随机 theta 方法的均方收敛率

本文重温了著名的随机 theta 方法 (STM)，用于具有非全局 Lipschitz 漂移和扩散系数的随机微分方程 (SDE)。在域 $$D \subset {{\mathbb {R}}}^d, d \in {{\mathbb {N}}}$$ D ⊂ R d , d ∈ N 中的耦合单调性条件下，我们提出一种使用方法参数 $$\theta \in [\tfrac{1}{2}, 1]$$ θ ∈ [ 1 2 , 1 ] 实现 STM 均方误差上限的新方法，仅涉及与精确的求解过程。这使我们能够轻松恢复所考虑方案的均方收敛率，而无需对数值近似值进行先验的高阶矩估计。作为误差界限的应用，我们在进一步的全局多项式增长条件下推导出由三种噪声驱动的 SDE 的 STM 的均方收敛率。特别是，误差界限用于分析具有小噪声的 SDE 的近似值。结果表明，随机梯形公式比其他 STM 具有更好的收敛性能。此外，我们将 STM 应用于 Ait-Sahalia 型利率模型，取值域 $$D = ( 0, \infty )$$ D = ( 0 , ∞ ) ，并成功识别出一阶收敛速度-即使在一般临界情况下，对于 $$\theta \in [\tfrac{1}{2}, 1]$$ θ ∈ [ 1 2 , 1 ] 的 STM 也有一半。这填补了 Szpruch 等人留下的空白。(BIT Numer Math 51(2):405–425, 2011)，证明了后向欧拉方法的强收敛性，但没有揭示收敛速度，