Let $X = (X_t: t \in [0,T])$ be a stochastic process satisfying a CIR model $$ dX_t = \beta (X_t - \gamma) dt + \sigma\sqrt{X_t} dB_t, $$ where $B_t$ is a standard Brownian motion, $\beta$ is a negative constant, $\gamma, \sigma$ are positive constants. In order for the SDE to make sense, assume that $X_t > 0$ for all $t \in [0,T]$.
Consider following two ways to simulate the model based on discretization of $t$ with Ito-Taylor expansion:
- the Euler scheme: $$ X_{t + \Delta} \approx X_t + \beta(X_t - \gamma)\Delta + \sigma \sqrt{X_t} Z \Delta, $$ where $Z$ is $N(0, 1)$ Gaussian variable.
- the Milstein scheme $$ X_{t + \Delta} \approx X_t + \beta(X_t - \gamma)\Delta + \sigma \sqrt{X_t}Z\sqrt{\Delta} + \frac{1}{4}\sigma^2 \Delta (Z^2-1) $$ where $Z$ is $N(0, 1)$ Gaussian variable.
I was wondering why these two schemes have a positive probability of generating negative values of $X_t$ and therefore cannot be used without suitable modifications?
References (book, tutorial and/or paper) will be helpful too!
Thanks and regards!