In this paper we study the rate of convergence of a symmetrized version ofthe Milstein scheme applied to the solution of the one dimensional SDE $$X_t =x_0 + \int_{0}^t{b(X_s)ds}+\int_{0}^t{\sigma |X_s|^\alpha dW_s},\;x_0>0,\;\sigma>0,\; \alpha\in[\tfrac{1}{2},1).$$ Assuming $b(0)/\sigma^2$ bigenough, and $b$ smooth, we prove a strong rate of convergence of order one,recovering the classical result of Milstein for SDEs with smooth diffusioncoefficient. In contrast with other recent results, our proof does not relieson Lamperti transformation, and it can be applied to a wide class of driftfunctions. On the downside, our hypothesis on the critical parameter value$b(0)/\sigma^2$ is more restrictive than others available in the literature.Some numerical experiments and comparison with various other schemes complementour theoretical analysis that also applies for the simple projected Milsteinscheme with same convergence rate.
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