The present work is devoted to strong approximations of a generalized Ait-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. To the best of our knowledge, this is the first paper to propose an unconditionally positivity preserving explicit scheme with order 1/2 of mean-square convergence for the model. Numerical experiments are finally provided to confirm the theoretical findings.

翻译：本文致力于数学金融中广义Ait-Sahalia模型的强逼近研究。该模型的数值研究面临若干本质困难：漂移项在原点处爆破、高度非线性的漂移与扩散系数以及保正性约束。本文提出一种新型显式Euler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。本文还证明了该格式在非临界与一般临界情形下均具有0.5阶均方收敛率。本研究的动机源于多层级蒙特卡洛模拟对该模型的应用需求——该方法要求均方收敛率，且在大离散时间步长下尤其需要保持正性。据我们所知，本文首次为上述模型提出了兼具无条件保正性与1/2阶均方收敛率的显式格式。最后通过数值实验验证了理论结果。