The PSP uses a custom Linux using the 2.6.x Kernel, and there is a firmware that'll run most full linux builds from a UMD (as they hold about 1.8GB) with all the features you'd expect from the desktop-pc versions.
While they'll have trouble running some aspects of DBP, forget what the current build of Cedega/Wine that is capable of DirectX9.0c it should work for the most part without many problems.
The real trouble comes with setting Linux up to run them really.
Ubuntu solves most of these for people without experience but you'll run into some other issues.
Hell failing that, you could just use ReactOS that'll also run as a boot-time that supports most of Windows 2000/XP features (including DirectX)
It's possibly, but real question is how much do you actually want to do it. Personally I'd suggest middleware that actually supports the PSP though; like GameBryo or Renderware.