In the study of stochastic processes, a stochastic process is adapted (also referred to as a non-anticipating or non-anticipative process) if information about the value of the process at a given time is available at that same time. An informal interpretation[1] is that X is adapted if and only if, for every realisation and every n, Xn is known at time n. The concept of an adapted process is essential, for instance, in the definition of the Itō integral, which only makes sense if the integrand is an adapted process.

Definition

Let

The process is said to be adapted to the filtration if the random variable is a -measurable function for each .[2]

Examples

Consider a stochastic process X : [0, T] × Ω → R, and equip the real line R with its usual Borel sigma algebra generated by the open sets.

See also

References

  1. ^ Wiliams, David (1979). "II.25". Diffusions, Markov Processes and Martingales: Foundations. Vol. 1. Wiley. ISBN 0-471-99705-6.
  2. ^ Øksendal, Bernt (2003). Stochastic Differential Equations. Springer. p. 25. ISBN 978-3-540-04758-2.