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Given efficient markets, asset prices should be unpredictable in the sense that any upcoming returns are uncorrelated with current or past returns. Hence for traded assets the price should follow something more similar to a GBM than an O-U process. However, many financial metrics are not prices; for example interest rates or volatility. O-U processes may ...


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The two processes are not pathwise equal. Here is a simulation (sample path) of the two processes $(t-\tau)dW_{\tau}$ and $W_\tau d\tau$: Note that both processes have the same value at the final time.


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If $X_t$ is square integrable, then the integral \begin{align*} \int_0^t X_{\tau} dW_{\tau} \end{align*} is a martingale. Here, the integrand $X_{\tau}$ does not depend on the integral limit $t$. However, in your case, the integrand, $t-\tau$, depends on $t$, then the condition for the martingality of the integral fails.



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