On explicit order 1.5 approximations with varying coefficients: The case of super-linear diffusion coefficients

A conjecture appears in Kumar and Sabanis (2016), in the form of a remark, where it is stated that it is possible to construct, in a specified way, any high order explicit numerical schemes to approximate the solutions of SDEs with superlinear coefficients. We answer this conjecture to the positive for the case of order 1.5 approximations and show that the suggested methodology works. Moreover, we explore the case of having Hölder continuous derivatives for the diffusion coefficients.