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 knowable[clarification needed] at a prior time. The predictable processes form the smallest class[clarification needed] that is closed under taking limits of sequences and contains all adapted left-continuous processes[clarification needed].

Mathematical definition[edit]

Discrete-time process[edit]

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[edit]

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[edit]

See also[edit]

References[edit]

  1. ^ van Zanten, Harry (November 8, 2004). "An Introduction to Stochastic Processes in Continuous Time" (pdf). Retrieved October 14, 2011. 
  2. ^ "Predictable processes: properties" (pdf). Retrieved October 15, 2011. [dead link]