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.

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 &sigma;-algebra $$\mathcal{F}_n$$ for each n.

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 &sigma;-algebra generated by all left-continuous adapted processes. This σ-algebra is also called the predictable σ-algebra.

Examples

 * Every deterministic process is a predictable process.
 * Every continuous-time adapted process that is left continuous is obviously a predictable process.