I am interested in ito processes (mostly ito diffusions) that "die" with some probability over time. E.g. A half life $\tau$ can be specified such that the expected lifetime of the process is $\tau$.
A special case I am interested in is where the probably of death is spatially (but not temporally) varying.
Specifically:
What are such processes called and what, if any, are some interesting studies on them?
Is there some computationally efficient procedure for simulating these?
Edit (6/16/17): it seems like one idea is to consider random upper limits of integration i.e. $$ X_t = \int_0^\tau \mu(X_s)ds + \int_0^\tau\sigma(X_s)ds $$ where $\tau$ is a random variable. This is obviously simple if $\tau$ is independent of $X_t$, but I am not sure what to do if I want to make it depend on the "history" of the process i.e. $\tau(X_s)$, so that if the process starts wandering in a particular area it is more likely to die early, for instance.
There does not seems to be much in the way of "random upper limits" for Ito processes (e.g. here and here mention it).