Existence and approximation of strong solutions of SDEs with fractional diffusion coefficients

In stochastic financial and biological models, the diffusion coefficients often involve the terms \begin{document}$ \sqrt{|x|} $\end{document} and \begin{document}$ \sqrt{|x(1-x)|} $\end{document} , or more general \begin{document}$ |x|^{r} $\end{document} and \begin{document}$ |x(1-x)|^r $\end{document} for \begin{document}$ r $\end{document} \begin{document}$ \in $\end{document} \begin{document}$ (0, 1) $\end{document} . These coefficients do not satisfy the local Lipschitz condition, which implies that the existence and uniqueness of the solution cannot be obtained by the standard conditions. This paper establishes the existence and uniqueness of the strong solution and the strong convergence of the Euler-Maruyama approximations under certain conditions for systems of stochastic differential equations for which one component has such a diffusion coefficient with \begin{document}$ r $\end{document} \begin{document}$ \in $\end{document} \begin{document}$ [1/2, 1) $\end{document} .