> 0  
\\
dV_t &=  c_1 V_t( c_2 -  V_t) \, dt +  c_3 V_t^{3/2}  \, dW^{(2)}_t, & \quad V_0=v_0>  0
$$ where $c_1,c_2,c_3 >0$, see e.g. [@heston_2].\
Motivated by these and other examples, the investigation of numerical methods for SDEs with non-Lipschitz coefficients has been an active field of research in recent years. This article, provides an overview of the new developments using the above equations as illustrative examples and discussing, in particular, Euler-type schemes. For some of the above equations exact simulation methods exist, see e.g. [@bk; @gm] and also [@roberts] for a class of one-dimensional equations, which are superior for the simulation of the SDEs at a single or a few time points. However, if a full sample path of the SDE has to be simulated or if the SDEs under consideration are part of a larger SDE system, then discretization schemes are typically more efficient.

Pathwise Convergence Rates of the Euler Scheme and general Itô-Taylor Methods
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The pathwise error criteria are very robust with respect to the global Lipschitz assumption. One of the simplest approximation schemes for equation (\[itosde\]) is the Euler scheme $$\begin{aligned}
\overline{X}_{t_{k+1}} &= \overline{X}_{t_{k}}  + a(\overline{X}_{t_{k}}) \Delta + \sum_{j=1}^m b_j (\overline{X}_{t_{k}}) \Delta_k 