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Got this question from my homework. I think if past returns are keep raising then current return should also be positive, but the answer is it's not related to past returns, why? I tried to ask chatgpt and it gave a function of $r = μ + σ^2/2$ but it couldn't explain why μ is not related to past returns. I think $μ$ can be the mean value of past returns and would connect current stock return to past stock returns.

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  • $\begingroup$ I think if past returns are [positive] then current return should also be positive. Generally that is not what statisticians who studied stock returns have found. $\endgroup$
    – nbbo2
    Apr 28, 2023 at 13:46

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If the returns follow Geometric Brownian Motion model then by definition the returns are independent of past returns. You are right in that returns are affected by past returns but there are models that assume Brownian motion for returns. For example, if you are doing some high-frequency or working in market microstructure models assuming Brownian motion will still be useful since looking at price moves in milliseconds looks very much random.

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