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

Continuous-time process

Given a filtered probability space $(\mathrm{\Omega },{ℱ},({{ℱ}}_t{)}_{t\ge 0},\mathbb{\mathbb{P}})$, then a continuous-time stochastic process $(X_t{)}_{t\ge 0}$ is predictable if $X$, considered as a mapping from $\mathrm{\Omega }\times {\mathbb{ℝ}}_{+}$, 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