Consider an Ornstein-Uhlenbeck stochastic process $(r_t)_{t\geq 0}$ given through the SDE: $$dr_t=\kappa \left( \mu-r_t\right)dt+\sigma dW_t,$$ where $r_0, \kappa, \mu$ and $\sigma$ are given positive constants.
It is known that it is a Gaussian process in a sense that $$r_t \sim \mathcal{N}\left(e^{-\kappa t} \left( r_0 -\mu \right)+\mu , \frac{\sigma^2}{2 \kappa} \left( 1-e^{-2\kappa t} \right)\right).$$
The algorithm for its simulation given as $$ r(t_{i+1})=e^{-\kappa \left( t_{i+1}-t_i\right)} \left( r(t_i)-\mu\right)+\mu +\sigma \sqrt{\frac{1}{2 \kappa}\left( 1-e^{-2 \kappa \left(t_{i+1}-t_t \right)} \right)}Z_{i+1}, $$ where $Z_{i+1}$ is a standard normal, is an exact simulation in a sense that the distribution of the $r(t_1), \dots, r(t_n)$ it produces is precisely that of the process at times $t_1, \dots, t_n$ for the same value of $r_0$.
There is something that's kinda puzzling to me about this, and don't know how to state it mathematically precisely, so apologies in advance.
Namely, if we assume that at some given time $t$ we have $$\mu-r_t>0,$$ i.e. $$ \mu>r_t,$$ looking at the SDE it's obvious that the drift term is positive, and the bigger the parameter $\kappa$, the bigger the drift term $\kappa (\mu-r_t)$ is, thus implying that the probability of $r_{t+\Delta t}>\mu$ is increasing with $\kappa$.
On the other hand, if we look at the simulation algorithm when $\mu-r_t>0$, if $\kappa$ is increasing the (negative) term $e^{-\kappa \Delta t} \left( r_t-\mu\right)$ is getting closer to zero from the left, so
$$ e^{-\kappa \Delta t} \left( r_t-\mu\right)+\mu \to \mu_-, $$ but also the std of the random variable $\sigma \sqrt{\frac{1}{2 \kappa}\left( 1-e^{-2 \kappa \Delta t} \right)}Z_{i+1}$ is getting smaller, so the probability of $r_{t+\Delta t}>\mu$ is I guess also kinda increasing with $\kappa$, but on the intuitive level it seems to me like it's not the same rate of increase? Like, when considering big $\kappa$ and the SDE - it looks like the probability of $r_t>\mu$ is quite large, but looking at the algorithm the bigger $\kappa$ will make us closer to the mean but we won't cross it with the drift part, and then the stochastic part is getting "less and less" stochastic, so there's less chance we're gonna cross $\mu$ actually.
My question is kinda all over the place, apologies for that. Would greatly appreciate any insights on this. Thanks!
So, $r_t$ is below the mean, and the SDE is saying that it should be increasing, ie., moving towards the mean from the left. This is consistent with what you've written about the simulation/solution.