\qquad k=0, 1, \ldots\end{aligned}$$ was studied in [@hm]. Both approaches extend the mapping $[0, \infty) \ni x \mapsto \sqrt{x} \in [0, \infty)$ suitably to negative values of $x$. For the CIR process this idea was taken further by Lord, Koekkoek & van Dijk [@lkd], who also proposed modifications of the drift coefficient for negative values of the state space.\

\[cir\_scen\_def\]

The following table shows the average number of negative steps per path for the above Euler approximations of the CIR process. Scenario I (taken from [@AS_K]), corresponds to the parameters $$x_0 = 0.05 , \qquad \kappa =5.07 , \qquad \lambda = 0.0457, \qquad \theta = 0.48, \qquad T=5$$ while Scenario II (taken from [@bk]) uses $$x_0=0.09, \qquad \kappa = 2, \qquad \lambda =0.09, \qquad \theta =1, \qquad T=5.$$ The stepsize for the Euler schemes is given by $\Delta =T/n$ with $n=512$.

   average negative steps of / for   Scenario I   Scenario II
  --------------------------------- ------------ -------------
            Euler scheme               0.9141       64.8611
            Euler scheme               1.0590       74.5017

The empirical frequency of negative paths is 0.4913 in Scenario I and 0.9990 in Scenario II. These results were obtained by a Monte Carlo simulation with $N=10^6$ repetition. They clearly indicate that the Euler scheme has a tendency 