So I have a set of time series data that follows (at least assumed) GBM but there are missing data. To interpolate, I'm thinking about simulation and create some sample paths. I have two options in mind and would like to know which one makes more sense.
(1) Estimate the parameters of the GBM using available data then interpolate with those parameters (GBM model).
(2) Modify the Brownian bridge model so that the bridge follows a GBM but then the parameters (mu and sigma) will be different from those in (1). E.g. the mu is log(BT/B0)/T+sigma^2/2 with B0 and BT the start and end of the bridge and T the length of increments.