I'm considering model patent right transfer as a survival process. That is, when a patent is transferred, it is considered death; if not, it's survival.
I'm not using Logitic regression because I want to know whether some policy will speed up the patent transfer process, and also we don't know whether some patent will be transferred yet.

But the problem is that only a small share of patents are transferred (about 10%), and most patent will never be transfer in its patent life. Could that be a problem for using Cox regression? If I remove these never transferred patents and only keep the ones that eventually transferred, would that be a problem?

My questions are basically:

1. Can I use Cox regression when a sample with a large proportion of observations never die?
2. Would it problematic if I conduct Cox regression on only those that eventually die?

Thank you for your reply!