On explicit order 1.5 approximations with varying coefficients: the case of super-linear diffusion coefficients
Provides theoretical validation for high-order explicit schemes in stochastic differential equations with superlinear coefficients, addressing a gap for numerical analysts.
The paper confirms a conjecture about constructing explicit order 1.5 numerical schemes for SDEs with superlinear diffusion coefficients, and extends the result to Hölder continuous derivatives.
A conjecture appears in \cite{milsteinscheme}, 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 affirmatively 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.