广义 Ait-Sahalia 模型的无条件保正显式 Euler 型格式
Unconditionally positivity-preserving explicit Euler-type schemes for a generalized Ait-Sahalia model
AI总结：
针对广义 Aït-Sahalia 模型在原点发散的漂移项及保正性难题，提出一种无条件保正的显式 Euler 型格式，在非临界与一般临界情形下均实现 0.5 阶均方收敛率，为 MLMC 模拟提供理论依据。
摘要：
本文致力于研究源于数学金融的广义 Aït-Sahalia 模型的强近似。由于原点处发散的漂移项、高度非线性的漂移和扩散系数以及保正性要求，该模型的数值研究面临本质困难。本文提出了一种新型显式 Euler 型格式，它易于实现，且能无条件地（即对任意时间步长 $h >0$）保持原始模型的正性。在非临界和一般临界情形下，所提格式均获得了 0.5 阶的均方收敛率。本研究的动机在于为该模型的 Multilevel Monte Carlo (MLMC) 模拟提供理论依据，此类模拟要求具备均方收敛率，且对于较大的离散时间步长，保正性尤为理想。最后，提供了数值实验以验证理论发现。
英文摘要：
The present work is devoted to strong approximations of a generalized Aït-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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
