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0

You are right, the mean is going to be $S_0 e^{\mu t}$. You may want to increase the number of simulations by the way, 1,000 ain’t that many. Since you know $(S_t)$ analytically in closed form, simulating and averaging does not really provide you with any further information. We know all moments of $(S_t)$ in closed-form anyway and can compute probabilities ...


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Let's try a simple approach, ignoring the difference between sample and population variance, and assuming the process is just the standard brownian - with no drift and sigma term. Generalisation should be easy. We define a process Y as equal to standard brownian, but we are assuming finite sampling with difference between two observations equal to $\Delta t$...


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