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.^{[clarification needed]}

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Mathematical definition

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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]}

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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]}
This σ-algebra is also called the **predictable σ-algebra**.