  We study the numerical approximation of numerous processes, solutions of
nonlinear stochastic differential equations, that appear in various
applications such as financial mathematics and population dynamics. Between the
investigated models are the CIR process, also known as the square root process,
the constant elasticity of variance process CEV, the Heston $3/2$-model, the
A\"it-Sahalia model and the Wright-Fisher model. We propose a version of the
semi-discrete method, which we call Lamperti semi-discrete (LSD) method. The
LSD method is domain preserving and seems to converge strongly to the solution
process with order $1$ and no extra restrictions on the parameters or the
step-size.
