  We are interested in strong approximations of one-dimensional SDEs which have
non-Lipschitz coefficients and which take values in a domain. Under a set of
general assumptions we derive an implicit scheme that preserves the domain of
the SDEs and is strongly convergent with rate one. Moreover, we show that this
general result can be applied to many SDEs we encounter in mathematical finance
and bio-mathematics. We will demonstrate flexibility of our approach by
analysing classical examples of SDEs with sublinear coefficients (CIR, CEV
models and Wright-Fisher diffusion) and also with superlinear coefficients
(3/2-volatility, Ait-Sahalia model).
  Our goal is to justify an efficient Multi-Level Monte Carlo (MLMC) method for
a rich family of SDEs, which relies on good strong convergence properties.
