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In its simplest form, the difference is "variance drag", ie how volatility itself affects your mu above. Imagine a series of random returns of +50% versus -50% with a 50% probability of each. Evidently this will have a mu of zero, and a sigma of 0.5. But if prices halve and appreciate by 50% with equal probability, they will clearly decline over ...


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If I am understanding your question correctly, maybe you can simulate two correlated GBM's and then apply the lag manually afterward.


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