I have run a simulation of a geometric brownian motion. The simulation runs from $t=0$ to $T=1000$. I generate $10000$ paths. For every moment for $t=1,2,3,\ldots, 100$ I calculate the $90$th percentile of the simulated paths. When I plot these points, depending on the used parameters, the curve that is produced is concave or convex. Please check the figure for an example.
When $\sigma$ is large relative to $\mu$, the curve of the $90$th percentile is concave (as above), and when $\sigma$ is small relative to $\mu$ the curve is convex. Is there any analytic formula for the distribution of percentiles of the GBM between two time instances? Also, is there an analytic or heuristic way to predict the shape of the 90th percentile (or any other percentile)? Thanks for the help.