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：本文旨在设计并分析一种高效的强一阶格式，用于数学金融中出现的广义Aït-Sahalia型模型，该模型定义在正域$(0, \infty)$上，其扩散项具有超线性增长，且漂移项高度非线性并在原点处发散。该模型复杂的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。通过在项$\alpha_{-1} x^{-1}$中引入隐式性，并在递归中使用修正映射$\Phi_h$，我们为所考虑的模型开发了一类新颖的显式且无条件保正（即对于任意步长$h>0$）的Milstein型格式。在非临界和一般临界情形下，我们引入了一种新方法来分析新格式的均方误差界，该方法不依赖于数值逼近的先验高阶矩有界性。所提出的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罗方法的最优复杂度。最后提供数值实验以验证理论结果。