Let's suppose the asset price process follows a Geometric Brownian motion $S_t \sim GBM(\mu, \sigma),\,t\ge 0$, and define the two process: $$ \begin{align} \text{MSF}_t &:= \max_{\tau\in[0,t]} S_\tau\\ \text{MDD_Ratio}_t &: = \max_{\tau\in[0,t]} ((\text{MSF}_\tau- S_\tau) / \text{MSF}_\tau) \end{align} $$ where MSF means "maximum so far" and MDD means maximum drawdown. As $S_t$ is forever positive, we see that MDD_Ratio is always well defined.
Is there any research done on the distribution, or at least the mean value of MDD_Ratio?
For what I saw, most research literature on DD seems to focus on the max drawdown value, i.e., $\max_{0\le t\le T}(\text{MSF}_t - S_t)$ (and most of the time the asset price process is assumed to follow a BM instead of a GBM), instead of maximum drawdown ratio whch IMHO is much commoner in practice.