Changing Measure to Simplify an Optimal Stopping Model

Summary

The document presents a question about a paper on deciding when to sell an asset whose price trend may be increasing or decreasing. The author is trying to understand a change of measure that is intended to simplify the model: the transformed process should become a geometric Brownian motion without explicitly carrying the hidden trend probability or odds ratio as state variables.

The question sketches the algebra. It relates the Brownian motion under the original measure to one under a new measure through a drift adjustment, substitutes that relation into the asset process, and chooses the adjustment to cancel a term involving the probability of the trend state. The point of confusion is why the paper’s drift for the transformed process contains an additional volatility factor. The document provides no answer, derivation from the paper, or validation of the proposed adjustment, so it identifies a modeling issue rather than resolving it. Readers would need the paper’s precise definitions and measure-change convention to determine whether the extra factor is correct.

Key ideas

• The cited optimal stopping model seeks to reduce state dimensionality by removing trend probabilities from the price process.
• A change of measure can shift drift terms while representing the process with a Brownian motion under a new measure.
• The question derives a drift adjustment intended to cancel dependence on the probability of the trend state.
• The document leaves unresolved why the paper’s transformed drift includes an extra volatility factor.

Tags

# optimal stopping time problem


# optimal stopping time problem












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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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.