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The first the solution to: $$dS_t = S_t\left[\mu dt +\sigma dW_t\right]$$ The second is the solution to: $$dS_t = S_t\left[\left(\mu -\frac{\sigma^2}{2}\right)dt + \sigma dW_t\right]$$ The difference is that the first one is a martingale when $\mu$ is equal to zero while the second one is not:  \mathbb{E}[S_0 exp(\sigma W_t)]= ...

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In the Ljung-Box test, the null hypothesis is: $H_0$: The data are independently distributed So, your p-values of 0 indeed indicate that you should reject the null hypothesis, but it means that your data is not independently distributed, and in particular that there is some significant autocorrelation in the process. This is obviously the case, because ...

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I would confirm it. For time series forecasting, one can use 3 versions of random walk: RW model 1 (basic geometric random walk): stock returns in different periods are statistically independent (uncorrelated) and identically distributed (constant volatility) RW model 2: stock returns in different periods are statistically independent bot not identically ...

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The approach of reflecting is expensive, since the $d$-simplex has $d$ maximal faces, all of which have to be checked for intersection at each step. Additionally, if the random walk moves into a corner, the number of moves which have to be discarded can become very high. Depending on the configuration of the constraints this could well be your best solution. ...

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There are several reasons: Expected payoff = 0 I don't know why you selected 0.5 ATR and 2 ATR away from the market but let's go with it for a while. This means that you want to gain 2x while risking only 0.5x. For now let's assume that FX log-returns are normal. To bring it to a higher level, we can use a piece of Black Scholes formula, namely the ...

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