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.

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.

Examples

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

See also

• Adapted process
• Martingale

References