Yes, that sounds sensible but careful not to fixate on mu being a thing. I don't think you are but, just in case, recall this thread...
https://www.acted.co.uk/forums/index.php?threads/april-2014-question-3.19981/
and in particular the bit where I write...
"Yes, GBM and cts time logN model are the same model, we just need to be careful what we mean by "mu".
If we start with dSt = St (mu dt + sigma dZt ) and solve, we will get log St = log S0 + (mu - 0.5sigma^2) t + sigma Zt, which is the same as saying St ~ logN [log S0 + (mu - 0.5sigma^2) t , sigma^2 t ]
If we start with log St = log S0 + mu t + sigma Zt, which is the same as saying St ~ logN [log S0 + mu t , sigma^2 t ], this would trace back to dSt = St ((mu +0.5 sigma^2) dt + sigma dZt )
Try not to fixate on mu as a thing. It's just a letter of the Greek alphabet. I could say that mu - 0.5sigma^2 = alpha and then hand my solution to GBM to a friend as log St = log S0 + alpha t + sigma Zt. This friend might then decide that they don't like alpha and they're going to change it to mu before handing it to a third person - that third person might get confused about why the mu in the first representation is not the same mu as the one in the second representation. That's all that's going on here, but it's still the SAME model,
John"
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