Killed process

In probability theory &mdash; specifically, in stochastic analysis &mdash; a killed process is a stochastic process that is forced to assume an undefined or "killed" state at some (possibly random) time.

Definition
Let X : T &times; &Omega; &rarr; S be a stochastic process defined for "times" t in some ordered index set T, on a probability space (&Omega;, &Sigma;, P), and taking values in a measurable space S. Let &zeta; : &Omega; &rarr; T be a random time, referred to as the killing time. Then the killed process Y associated to X is defined by


 * $$Y_{t} = X_{t} \mbox{ for } t < \zeta,$$

and Yt is left undefined for t &ge; &zeta;. Alternatively, one may set Yt = c for t &ge; &zeta;, where c is a "coffin state" not in S.