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 $(\Omega ,{\mathcal {F}},({\mathcal {F}}_{n})_{{n\in {\mathbb {N}}}},{\mathbb {P}})$ , then a stochastic process $(X_{n})_{{n\in {\mathbb {N}}}}$ is predictable if $X_{{n+1}}$ is measurable with respect to the σ-algebra $\mathcal{F}_n$ for each n.^[1]

Continuous-time process

Given a filtered probability space $(\Omega ,{\mathcal {F}},({\mathcal {F}}_{t})_{{t\geq 0}},{\mathbb {P}})$ , then a continuous-time stochastic process $(X_{t})_{{t\geq 0}}$ is predictable if $X$ , considered as a mapping from $\Omega \times {\mathbb {R}}_{{+}}$ , 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.

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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