In probability theory

Probability theory

Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...

 — 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

Ordered set

In order theory in mathematics, a set with a binary relation R on its elements that is reflexive , antisymmetric and transitive is described as a partially ordered set or poset...

 index set

Index set

In mathematics, the elements of a set A may be indexed or labeled by means of a set J that is on that account called an index set...

 T, on a probability space

Probability space

In probability theory, a probability space or a probability triple is a mathematical construct that models a real-world process consisting of states that occur randomly. A probability space is constructed with a specific kind of situation or experiment in mind...

 (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 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.