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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

• Continuous-time process

• Examples

• See also

• References

Given a filtered probability space ( Ω Ω , F , ( F n ) n ∈ ∈ N , P ) {\displaystyle (\Omega ,{\mathcal {F}},({\mathcal {F}}_{n})_{n\in \mathbb {N} },\mathbb {P} )} , then a stochastic process ( X n ) n ∈ ∈ N {\displaystyle (X_{n})_{n\in \mathbb {N} }} is predictable if X n + 1 {\displaystyle X_{n+1}} is measurable with respect to the σ-algebra F n {\displaystyle {\mathcal {F}}_{n}} for each n.cite-ref-zanten-1-0[1]

Given a filtered probability space ( Ω Ω , F , ( F t ) t ≥ ≥ 0 , P ) {\displaystyle (\Omega ,{\mathcal {F}},({\mathcal {F}}_{t})_{t\geq 0},\mathbb {P} )} , then a continuous-time stochastic process ( X t ) t ≥ ≥ 0 {\displaystyle (X_{t})_{t\geq 0}} is predictable if X {\displaystyle X} , considered as a mapping from Ω Ω × × R + {\displaystyle \Omega \times \mathbb {R} _{+}} , is measurable with respect to the σ-algebra generated by all left-continuous adapted processes.cite-ref-2[2] This σ-algebra is also called the predictable σ-algebra.

• Every deterministic process is a predictable process.

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

• Adapted process

• Martingale

cite-note-zanten-11. ↑ citerefvan-zanten2004van 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.

cite-note-22. ↑ "Predictable processes: properties" (PDF). Archived from the original (pdf) on March 31, 2012. Retrieved October 15, 2011.