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by definition once you remove the trend and other volatility clustering patterns, you are left with white noise, without correlation whatsoever

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Part 1: deriving the drift of the stock price process under the stock Numeraire. Under the risk-neutral measure, the process for $S_t$ is as follows: $$S_t = S_0 + \int_{h=t_0}^{h=t}rS_h dh + \int_{h=t_0}^{h=t}\sigma S_h dW_h = \\ = S_0exp\left[ (r-0.5 \sigma^2)t+\sigma W(t) \right]$$ In the above model, the Numeraire is $N(t)=e^{rt}$ with $N(t_0):=1$. ...

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I tried a MonteCarlo simulation with 10000 iterations and it seems to confirm a probability of about 0.0033 However I wonder if we are interpreting the problem correctly, we are assuming infinite capital, perhaps we should take into account the gambling has to stop when you hit zero capital? Any other issues I may have missed? With the vig taken into account ...

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