Say I have a time series $S_K$ for monthly asset prices for the last 30 years. I want to run a monte carlo simulation using geometric brownian motion
$$S_t = S_0\exp\left(\left(\mu - \frac{\sigma^2}{2}\right)t + \sigma W_t\right)$$
In my monte carlo simulation, I plan to use a time increment $dt=\frac{1}{12}$ to simulate 1 month increments.
What is the mean $\mu$ and volatility $\sigma$ that should be used in the calculation? Intuitively, using the long term (30 year) mean and standard deviation seem incorrect as the simulation will have 1 month time steps, so I'm unsure what values to use.