I am assuming that an observed value $\hat Y_t$ is simulated from a known true value $Y_t$. I am also assuming that there is a log-normally distributed error that displays an AR1 process thus $\hat Y_t = Y_t * e^{X_t}$ , and $X_t = c + \rho*X_{t-1} + \epsilon_t$, where $\epsilon_t \sim N(0,\sigma^2)$
I took the natural log of the observed value minus the natural log of the true value and fit an AR(1) process to these errors ($X_t = ln(\hat Y_t) - ln(Y_t)$ ) using the arima function in R to estimate the coefficients for $c , \rho$ and the variance of $\epsilon_t$.
I now want to simulate a process that has the same variance and mean of the observed process. However, I am concerned that by using the estimated coefficients to simulate the AR(1) process and exponentiating the resulting random variables will result in a mean biased process
How would I simulate values for $\hat Y_t$ given the estimates so that the simulated values have the same mean and variance as the observed data?