Unconditionally positivity-preserving approximations of the Aït-Sahalia type model: Explicit Milstein-type schemes

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 \(\alpha _{-1} x^{-1}\) 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.