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：本文旨在为数学金融中出现的广义Aït-Sahalia型模型设计与分析高效的一阶强格式，该模型定义于正域$(0, \infty)$，其扩散项具有超线性增长特性，且漂移项在原点处具有高度非线性奇异性。模型的这种复杂结构不可避免地给时间离散化格式的构造与收敛性分析带来了本质性困难。通过在递归式中对项$\alpha_{-1} x^{-1}$引入隐式处理并结合校正映射$\Phi_h$，我们为底层模型发展了一类新颖的显式且无条件保正（即对任意步长$h>0$均成立）的Milstein型格式。在非临界与一般临界情形下，我们提出了一种新颖的方法来分析新格式的均方误差界，而无需依赖数值近似解的高阶矩先验估计。所提格式达到了理论预期的一阶均方收敛阶。上述理论保证可用于验证多层蒙特卡洛方法的最优计算复杂度。最后通过数值实验验证了理论结果。