Strong convergence of the symmetrized Milstein scheme for some CEV-like SDEs

In this paper we study the rate of convergence of a symmetrized version of the Milstein scheme applied to the solution of the one dimensional SDE $$X_t=x_0+{\int }_0^{t}b(X_s)ds+{\int }_0^t\sigma \mathrm{\mid }{X}_s{\mathrm{\mid }}^{\alpha }d{W}_s,x_0>0,\sigma >0,\alpha \in [\frac{1}{2},1)\mathrm{.}$$ Assuming $b(0)\mathrm{/}{\sigma }^2$ big enough, and $b$ smooth, we prove a strong rate of convergence of order one, recovering the classical result of Milstein for SDEs with smooth diffusion coefficient. In contrast with other recent results, our proof does not relies on Lamperti transformation, and it can be applied to a wide class of drift functions. On the downside, our hypothesis on the critical parameter value $b(0)\mathrm{/}{\sigma }^2$ is more restrictive than others available in the literature. Some numerical experiments and comparison with various other schemes complement our theoretical analysis that also applies for the simple projected Milstein scheme with same convergence rate.