PlayStation Vita is backwards compatible with PSP games, but it won’t play UMDs. Sony Computer Entertainment Japan’s "solution" to the problem is selling a discounted PSN downloads. Users have to register their game for a UMD Passport before they can purchase a digital copy at a reduced price. Prices range from 500 yen (Shin Megami Tensei: Persona) to 1,500 yen (Shin Megami Tensei: Persona 3 Portable).
A list of 262 UMD Passport games is live. While that is a chunk of the PSP’s library, there are 606 PSP games available for purchase on Japan’s PlayStation Store. On closer inspection, it appears only some publishers are supporting Sony’s digital discount plan. Games from Atlus, Nippon Ichi, Acquire, Tecmo Koei, SNK Playmore, and Cyberfront are available. Titles from key publishers such as Square Enix, Namco Bandai, Konami, and Monster Hunter maker Capcom are not supporting the UMD Passport service yet.
As of now, Sony Computer Entertainment America and Europe have not announced a similar service.