Take for example this passage from "Monte Carlo Methods in Financial Engineering".
Is this a result of the risk neutral world or is this the real world drift as well? I've never seen the explicit distinction made and I don't see anything in the argument that would imply it only to hold for the risk neutral world. Is this something that can actually be observed in the real world? If it were wouldn't we be seeing consistently rising (falling) FX rates between for example the USD and EUR (do we?) and if so wouldn't we expect the interest rates to start converging to each other as a result (do we?)?
Put in another way, if we were to do a Monte Carlo value at risk simulation of the FX rate to capture the risk of FX derivatives, would we use this drift $\mu = r-r_f$? Or would we only use this drift if simulating the FX rate to, say, value a call option on the FX rate and use zero drift in the Value at Risk simulation (similar as to how we use r as the drift in equity option valuation and zero drift when performing a monte carlo value at risk simulation)?