\sigma, r, \rho > 0$, $i=-1, \ldots, 2$. Under certain conditions on the parameters (see [@SMHJ]), this SDE has a unique strong solution with values in $(0, \infty)$. Note that here the diffusion coefficient grows superlinearly for large values of $x$ while the drift coefficient has a singularity at $x=0$.

The Heston model [@heston], which is an asset price model with stochastic volatility, is another example for an SDE with non-Lipschitz coefficients. This SDE takes non-negative values only and contains square root coefficients: $$\begin{aligned}
dS_t &= \mu
S_t \, dt + \sqrt{V_t} S_t  \, \big( \sqrt{1-\rho^2} \, d W^{(1)}_t + \rho\, dW^{(2)}_t\big),&  \quad S_0=s_0  > 0   \\ 
dV_t &= \kappa( \lambda -  V_t) \, dt +  \theta \sqrt{V_t}  \, dW^{(2)}_t, & \quad V_0=v_0 > 0. $$ The parameters satisfy $\mu \in \mathbb{R}$, $\kappa, \lambda, \theta > 0$ and $\rho \in (-1,1)$. The second component of this SDE is the Cox-Ingersoll-Ross process, which is also used as a short rate model [@cox:1985].

Finally, the use of the inverse of the CIR process as volatility process leads to the so-called $3/2$-model $$\begin{aligned}
dS_t &= \mu
S_t \, dt + \sqrt{V_t} S_t   \, \big( \sqrt{1-\rho^2} \, d W^{(1)}_t + \rho\, dW^{(2)}_t\big),&  \quad S_0=s_0  