Predictable process

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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 , then a stochastic process is predictable if is measurable with respect to the σ-algebra for each n.[1]

Continuous-time process

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]

Examples

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See also

References


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