Predictable process

Summary

In stochastic analysis, a part of the mathematical theory of probability, a predictable process is a stochastic process whose value is knowable at a prior time. The predictable processes form the smallest class that is closed under taking limits of sequences and contains all adapted left-continuous processes.[clarification needed]

Mathematical definition edit

Discrete-time process edit

Given a filtered probability space  , then a stochastic process   is predictable if   is measurable with respect to the σ-algebra   for each n.[1]

Continuous-time process edit

Given a filtered probability space  , then a continuous-time stochastic process   is predictable if  , considered as a mapping from  , is measurable with respect to the σ-algebra generated by all left-continuous adapted processes.[2] This σ-algebra is also called the predictable σ-algebra.

Examples edit

  • Every deterministic process is a predictable process.[citation needed]
  • Every continuous-time adapted process that is left continuous is obviously a predictable process.[citation needed]

See also edit

References edit

  1. ^ van Zanten, Harry (November 8, 2004). "An Introduction to Stochastic Processes in Continuous Time" (PDF). Archived from the original (pdf) on April 6, 2012. Retrieved October 14, 2011.
  2. ^ "Predictable processes: properties" (PDF). Archived from the original (pdf) on March 31, 2012. Retrieved October 15, 2011.