Predictable process
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.
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.
- Every continuous-time adapted process that is left continuous is a predictable process.
See also
References
- ↑ van Zanten, Harry (November 8, 2004). "An Introduction to Stochastic Processes in Continuous Time" (pdf). Retrieved October 14, 2011.
- ↑ "Predictable processes: properties" (PDF). Archived from the original (pdf) on March 31, 2012. Retrieved October 15, 2011.
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