I'm currently reading a paper (The Optimal Stopping Time for Selling an Asset When It Is Uncertain Whether the Price Process Is Increasing or Decreasing, American Journal of Operations Research, March 2018) about an optimal stopping time problem.

My issue is that I'm unable to see why $\sigma$ was added in the drift term of the process $dZ_t$. From what was explained to me, what the authors want to do is to get to the expression $(6)$. Which has the advantage of being a simple geometric brownian motion where we no longer have to deal with $\pi_t$. The same goes to $\phi_t$, the odds ratio, reducing thus the dimensionality of the problem.

If we want to reverse engineer the definition of $Z_t$, we know that we have something in the form: $d\bar{W}_t=dZ_t+c_tdt$ where $dZ_t$ is a Brownian motion under another measure $Q$. This actually defines $Z_t$ as $dZ_t=-c_t dt+d \bar{W}_t$. Plugging this into $dX_t$ we obtain: $dX_t/X_t = \sigma[(-\omega*\pi_t + \mu_2/\sigma + c_t)dt + dZ_t]$ where $\omega = (\mu_2-\mu_1)/\sigma$. We can now set $c_t = \omega*\pi_t$ to get rid of the $\pi_t$ term.

However and for some reason the drift term of the $dZ_t$ process (unnumbered equation after equation (3)) contains an additional $\sigma$! Can anyone please explain why.



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