Killed process
{{Short description|Stochastic process that is forced to assume an undefined or "killed" state at some time}}
{{For|killed processes in computer operating systems|Process state#Terminated}}
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In probability theory — specifically, in stochastic analysis — 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 × Ω → S be a stochastic process defined for "times" t in some ordered index set T, on a probability space (Ω, Σ, P), and taking values in a measurable space S. Let ζ : Ω → T be a random time, referred to as the killing time. Then the killed process Y associated to X is defined by
:
and Yt is left undefined for t ≥ ζ. Alternatively, one may set Yt = c for t ≥ ζ, where c is a "coffin state" not in S.
See also
References
{{Reflist}}
- {{cite book
| last = Øksendal
| first = Bernt K.
| authorlink = Bernt Øksendal
| title = Stochastic Differential Equations: An Introduction with Applications
| edition = Sixth
| publisher=Springer
| location = Berlin
| year = 2003
| isbn = 3-540-04758-1
}} (See Section 8.2)