I'm working on a 50-years financial timeseries and I would like to simulate GBM paths from it. The first thing I'm supposed to do is to estimate the drift $\mu$ and the volatility $\sigma$ parameters.
What I've done so far is to come up with the estimate on the entire history, but I think it's quite a big approximation since the expected return and the volatility of a financial asset vary over time. Also, it is known that GBM doesn't work so well on long time interval.
I thought that I could have time-dependent deterministic parameters for $\mu = \mu(t)$ and $\sigma = \sigma(t)$ where I compute the values for every years and update the parameters of the GBM accordingly, but I'm not sure if it makes sense.