Optimal Stopping Time Problem

[Goldman Clarck]

Hi everyone, I'm reading a paper titled "The Optimal Stopping Time for Selling an Asset When It Is Uncertain Whether the Price Process Is Increasing or Decreasing When the Horizon Is Infinite," which can be found for free in the link below [1]. The problem I find 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, 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 an other measure Q. This actually defines Z_t as
dZ_t=-c_t dt+d bar{W}_t. Plugging this in to 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 contains an additional sigma! Can anyone please explain why?

[1] https://www.scirp.org/journal/paperinformation.aspx?paperid=82927#ref3