stochastic differential equations (SDEs) relies on the global Lipschitz assumption, i.e. the SDE coefficients satisfy $$\begin{aligned}
|a(x)-a(y)| + \sum_{j=1}^m |b_{j}(x)-b_j(y)| \leq L \cdot |x-y|, \qquad x,y \in \mathbb{R}^d            \end{aligned}$$ for some $L>0$. However, in many SDEs used for modelling in mathematical finance this assumption is violated, so the standard results (see [@KP; @M]) do not apply.\
The Constant Elasticity of Variance Model for asset prices [@cox:1975] , which was introduced by Cox in 1975, is given by the SDE $$\begin{aligned}
 dS_t=\mu S_t \, dt + \sigma S_t^{\gamma} \, dW_t, \qquad S_0=s_0>0\end{aligned}$$ where $\mu \in {\mathbb R}$, $\sigma  >0$ and $\gamma \in (0,1]$ and $W_t, t \geq 0,$ is a one-dimensional Brownian motion. For $\gamma =1$ this is the standard Black-Scholes model (i.e. a geometric Brownian motion), while for $\gamma \in (0,1)$ the diffusion coefficient of this SDE is clearly not globally Lipschitz continuous. This SDE has a unique strong solution if and only if $\gamma \in   [1/2,1]$ and takes values in $[0,\infty)$.

The Ait-Sahalia model and its generalization [@AS; @SMHJ], which are stochastic interest rate models, follow the dynamics $$\begin{aligned}
dX_t =  \big(\alpha_{-1} X_t^{-1} - \alpha_0 + \alpha_1 X_t - \alpha_2X_t^r \big)dt +\sigma X_t^{\rho} dW_t, \qquad X_0=x_0>0\end{aligned}$$ where $ \alpha_{i}, 