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. 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