The present work is devoted to strong approximations of a generalized A\"{i}t-Sahalia model arising from mathematical finance. The numerical study of the considered model faces essential difficulties caused by a drift that blows up at the origin, highly nonlinear drift and diffusion coefficients and positivity-preserving requirement. In this paper, a novel explicit Euler-type scheme is proposed, which is easily implementable and able to preserve positivity of the original model unconditionally, i.e., for any time step-size $h >0$. A mean-square convergence rate of order $0.5$ is also obtained for the proposed scheme in both non-critical and general critical cases. Our work is motivated by the need to justify the multi-level Monte Carlo (MLMC) simulations for the underlying model, where the rate of mean-square convergence is required and the preservation of positivity is desirable particularly for large discretization time steps. Numerical experiments are finally provided to confirm the theoretical findings.

翻译：本文致力于研究金融数学中广义Aït-Sahalia模型的强近似问题。该模型的数值研究面临三大本质困难：原点处发散的漂移项、高度非线性的漂移与扩散系数，以及保正性要求。本文提出了一种新型显式欧拉型格式，该格式易于实现且能无条件保持原模型的正性，即对任意时间步长$h >0$均成立。我们同时证明了该格式在非临界和一般临界情形下具有$0.5$阶均方收敛速度。本研究源于对模型进行多层蒙特卡洛（MLMC）模拟的需求：该方法需要均方收敛速率，且在大离散时间步长情况下特别需要保持正性。最后通过数值实验验证了理论结果。