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Predictable process

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Title: Predictable process  
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Subject: Doob decomposition theorem, Sigma-martingale, Adapted process, Predictability, Malliavin calculus
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Predictable process

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

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

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]


See also


  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. 

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