Unconditionally positivity-preserving approximations of the Ait-Sahalia type model: Explicit Milstein-type schemes
Yingsong Jiang, Ruishu Liu, Xiaojie Wang, Jinghua Zhuo
TL;DR
The paper addresses the challenge of simulating Aït-Sahalia-type SDEs in $(0,\infty)$ with non-globally Lipschitz coefficients, a drift that blows up at the origin, and a diffusion with superlinear growth. It introduces an explicit Milstein-type scheme augmented by a corrective mapping $\Phi_h$ and an implicit-positivity term, ensuring unconditional positivity and tractability. The authors prove mean-square convergence of order one in both the noncritical ($r+1>2\rho$) and critical ($r+1=2\rho$) regimes under mild parameter conditions (notably $\alpha_2/\sigma^2\ge 4r+\tfrac{1}{2}$ in the critical case), without relying on a priori high-order moment bounds, and discuss implications for efficient multilevel Monte Carlo simulation. Numerical experiments corroborate the theoretical results, demonstrating near-linear mean-square convergence and substantial performance gains over traditional schemes while preserving positivity for all step sizes $h>0$.
Abstract
The present article aims to design and analyze efficient first-order strong schemes for a generalized Aït-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 $α_{-1} x^{-1}$ and a corrective mapping $Φ_h$ 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.