and to the 3/2-model with $D=(0,\infty)^2$ and no further restrictions on the parameter.

To illustrate the above results consider Scenario I for the Cox-Ingersoll-Ross process. Figure \[figure\_cir\_path\] shows for two different sample paths $\omega  \in \Omega$ the maximum error in the discretization points, i.e. $$\sup_{k=0, \ldots, n} |X_{t_k}(\omega)-\overline{X}_{t_{k}}
(\omega)|,$$ of

-   the truncated Euler scheme

-   the symmetrized Euler scheme

-   the modified Milstein scheme with auxiliary functions $f(x)=\kappa(\lambda- x)$, $g(x)=0$, i.e. a truncated Milstein scheme.

To estimate the pathwise maximum error for the above approximation schemes the Cox-Ingersoll-Ross process have been discretized with a very small step size using scheme .

 

In Figure \[figure\_cir\_path\] log-log-coordinates are used, so the dots indicate the convergence orders $0.5$ and $1$. The pathwise convergence rates of all three approximation schemes are in good accordance with the theoretically predicted rates for moderate and small step sizes. For small step sizes both Euler schemes do not take negative values and hence coincide. Moreover, for small step sizes the Milstein scheme is superior due to its first order convergence.

$\diamond$

Numerical methods with pathwise convergence rates of high order are thus available also for SDEs with non-globally Lipschitz coefficients. However, while pathwise convergence rates are very 