Predictable process explained

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

(\Omega, l{F},( l{F} _{n)} _{n} , P ) , then a stochastic process (X _{n)} _{n} is predictable if X _{n+1} is measurable with respect to the σ-algebra l{F} _{n} for each n.^[1]

Continuous-time process

Given a filtered probability space

(\Omega, l{F},( l{F} _{t)} _{t} , P ) , then a continuous-time stochastic process (X _{t)} _{t} is predictable if X , considered as a mapping from \Omega x R _{+} , 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

• Every deterministic process is a predictable process.
• Every continuous-time adapted process that is left continuous is obviously a predictable process.

See also

• Adapted process
• Martingale

Notes and References

1. Web site: An Introduction to Stochastic Processes in Continuous Time. Harry. van Zanten. November 8, 2004. pdf. October 14, 2011 . https://web.archive.org/web/20120406084950/http://www.cs.vu.nl/~rmeester/onderwijs/stochastic_processes/sp_new.pdf . April 6, 2012 . dead.
2. Web site: Predictable processes: properties . pdf . October 15, 2011 . dead . https://web.archive.org/web/20120331074812/http://www.math.ku.dk/~jesper/teaching/b108/slides38.pdf . March 31, 2012 .