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.^{[clarification needed]}
Contents
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.
See also[edit]
References[edit]
- ^ 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.
- ^ "Predictable processes: properties" (PDF). Archived from the original (pdf) on March 31, 2012. Retrieved October 15, 2011.