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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 knowableTemplate:Clarify at a prior time. The predictable processes form the smallest classTemplate:Clarify that is closed under taking limits of sequences and contains all adapted left-continuous processesTemplate:Clarify.

Mathematical definition

Discrete-time process

Given a filtered probability space (Ω,,(n)n,), then a stochastic process (Xn)n is predictable if Xn+1 is measurable with respect to the σ-algebra n for each n.[1]

Continuous-time process

Given a filtered probability space (Ω,,(t)t0,), then a continuous-time stochastic process (Xt)t0 is predictable if X, considered as a mapping from Ω×+, is measurable with respect to the σ-algebra generated by all left-continuous adapted processes.[2]

Examples

See also

References

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