The present article aims to design and analyze efficient first-order strong schemes for a generalized A\"{i}t-Sahalia type model arising in mathematical finance and evolving in a positive domain $(0, \infty)$, which possesses a diffusion term with superlinear growth and a highly nonlinear drift that blows up at the origin. Such a complicated structure of the model unavoidably causes essential difficulties in the construction and convergence analysis of time discretizations. By incorporating implicitness in the term $\alpha_{-1} x^{-1}$ and a corrective mapping $\Phi_h$ in the recursion, we develop a novel class of explicit and unconditionally positivity-preserving (i.e., for any step-size $h>0$) Milstein-type schemes for the underlying model. In both non-critical and general critical cases, we introduce a novel approach to analyze mean-square error bounds of the novel schemes, without relying on a priori high-order moment bounds of the numerical approximations. The expected order-one mean-square convergence is attained for the proposed scheme. The above theoretical guarantee can be used to justify the optimal complexity of the Multilevel Monte Carlo method. Numerical experiments are finally provided to verify the theoretical findings.

翻译：本文旨在设计和分析一种高效的一阶强格式，用于数学金融领域中出现的广义Aït-Sahalia类型模型，该模型在正域$(0, \infty)$上演化，具有超线性增长的扩散项和在原点处爆炸的高度非线性漂移项。该模型的复杂结构不可避免地给时间离散化的构造和收敛性分析带来了本质困难。通过在递推关系中引入项$\alpha_{-1} x^{-1}$的隐式处理和校正映射$\Phi_h$，我们为底层模型开发了一类新型显式且无条件保正（即对任意步长$h>0$均成立）的Milstein型格式。在非临界和一般临界情形下，我们引入了一种无需依赖数值近似先验高阶矩界的新方法来分析新型格式的均方误差界。所提格式实现了预期的一阶均方收敛。上述理论保证可用于证明多层蒙特卡洛方法的最优复杂性。最后通过数值实验验证了理论发现。