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 [math]\displaystyle{ (\Omega,\mathcal{F},(\mathcal{F}_n)_{n \in \mathbb{N}},\mathbb{P}) }[/math], then a stochastic process [math]\displaystyle{ (X_n)_{n \in \mathbb{N}} }[/math] is predictable if [math]\displaystyle{ X_{n+1} }[/math] is measurable with respect to the σ-algebra [math]\displaystyle{ \mathcal{F}_n }[/math] for each n.

Continuous-time process

Given a filtered probability space [math]\displaystyle{ (\Omega,\mathcal{F},(\mathcal{F}_t)_{t \geq 0},\mathbb{P}) }[/math], then a continuous-time stochastic process [math]\displaystyle{ (X_t)_{t \geq 0} }[/math] is predictable if [math]\displaystyle{ X }[/math], considered as a mapping from [math]\displaystyle{ \Omega \times \mathbb{R}_{+} }[/math], is measurable with respect to the σ-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.

See also

• Adapted process
• Martingale