Predictable process

(Redirected from Predictable σ-algebra)

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

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Discrete-time process

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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

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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

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See also

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References

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  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.