Predictable process
In stochastic analysis, a part of the mathematical theory of probability, a predictable process is a stochastic process whose value is knowable[clarification needed] at a prior time. The predictable processes form the smallest class[clarification needed] that is closed under taking limits of sequences and contains all adapted left-continuous processes[clarification needed].
Contents
Mathematical definition
Discrete-time process
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
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]
Examples
- Every deterministic process is a predictable process.[citation needed]
- Every continuous-time adapted process that is left continuous is a predictable process.[citation needed]
See also
References
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