Question
What is the distribution of the running maximum (DRM) of the following SDE: $$ dV_{t} = (\mu -\kappa V_{t})dt + \sigma dW_{t} $$
Previous Effort
The SDE was found and slightly modified from here (Eq. 2.14, describing a trending Ornstein-Uhlenbeck Process). My SDE is modified so that the random walk is affected by drift $\kappa$ to bring it down to 0, as opposed to the trend line $\mu t$.
The solution found when the unmodified equation is integrated (Eq. 2.15) has 3 terms, two deterministic, and one stochastic. If an expression can be found for the DRM of the stochastic portion of Eq. 2.15 ($\sigma\int_{0}^{t}e^{\kappa(\mu-t)}dW_{\mu}$), I believe that a DRM can be found for the entire trajectory (basically the variance is the same as the variance of the stochastic term's DRM, and the expected value is just the expected value of the stochastic term's DRM plus the value of the remaining deterministic terms), which would naturally lead to the solution to my problem.
This document shows how the DRM is found for a Wiener process (pg. 6), but I'm not sure if the proof will apply to whatever the stochastic term of my trajectory ends up being.
Does this seem like the right path, and if so how can I get an expression for the DRM of the stochastic portion of Eq. 2.15 (as above)? Otherwise, are there any other ideas on where to find an expression for the DRM of my SDE? Any help is greatly appreciated!