Strange sign reversal of simulated series in stochastic simulation

My understanding of stochastic simulation with a periods option is that dynare takes a single draw of all endogenous variables based on the loglinearized decision processes. I see something strange when I look at the generated series. I have a multiplicative process, say x(t)=xbar^(1-lambda)x(t-1)^lamdaexp(epsilon) where lambda is the autocorrelation coefficient and epsilon is the Gaussian white noise. x(t) by construction can never be negative. But when I see the simulated series from dynare I see many times x(t) is negative.
Same thing happens if I estimate the model and look at the smoothed x series. This really baffles me. Can anyone please explain or refer to a source where it is explained how dynare exactly does the stochastic simulation?

See e.g.

I figured out that dynare actually linearises the multiplicative processs NOT loglinearises. This is what I see in the manual. Thus it also linearises exp(epsilon) which means that for a large shock x can become negative. I don’t see any way out keeping it positive except giving a small standard error.

4.13.3 First order approximation

The approximation has the stylized form:

where is the steady state value of and .

The coefficients of the decision rules are stored as follows:

Yes, but note that when using exp()-substitution you are doing a log-linearization. But in any case, linearization means that you cannot impose bounds.