In
probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...
— specifically, in
stochastic analysis
Stochastic calculus is a branch of mathematics that operates on stochastic processes. It allows a consistent theory of integration to be defined for integrals of stochastic processes with respect to stochastic processes. This field was created an ...
— 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
In mathematics, an index set is a set whose members label (or index) members of another set. For instance, if the elements of a set may be ''indexed'' or ''labeled'' by means of the elements of a set , then is an index set. The indexing consists ...
''T'', on a
probability space
In probability theory, a probability space or a probability triple (\Omega, \mathcal, P) is a mathematical construct that provides a formal model of a random process or "experiment". For example, one can define a probability space which models t ...
(Ω, Σ, P), and taking values in a
measurable space
In mathematics, a measurable space or Borel space is a basic object in measure theory. It consists of a set and a σ-algebra, which defines the subsets that will be measured.
Definition
Consider a set X and a σ-algebra \mathcal A on X. Then the ...
''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 ''Y''
''t'' is left undefined for ''t'' ≥ ''ζ''. Alternatively, one may set ''Y''
''t'' = ''c'' for ''t'' ≥ ''ζ'', where ''c'' is a "coffin state" not in ''S''.
See also
*
Stopped process In mathematics, a stopped process is a stochastic process that is forced to assume the same value after a prescribed (possibly random) time.
Definition
Let
* (\Omega, \mathcal, \mathbb) be a probability space;
* (\mathbb, \mathcal) be a measurable ...
References
* {{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)
Stochastic processes