{{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}} {{no footnotes|date=July 2024}} {{one source|date=July 2024}} {{Use dmy dates|date=July 2024}} 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
:<math>Y_{t} = X_{t} \mbox{ for } t < \zeta,</math>
and ''Y''<sub>''t''</sub> is left undefined for ''t'' ≥ ''ζ''. Alternatively, one may set ''Y''<sub>''t''</sub> = ''c'' for ''t'' ≥ ''ζ'', where ''c'' is a "coffin state" not in ''S''.
==See also== * Stopped process
==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)
Category:Stochastic processes