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 $(\Omega,\mathcal,(\mathcal_n)_,\mathbb)$, then a stochastic process $(X_n)_$ is ''predictable'' if $X_$ is measurable with respect to the σ-algebra $\mathcal_n$ for each ''n''.

Continuous-time process

Given a filtered probability space $(\Omega,\mathcal,(\mathcal_t)_,\mathbb)$, then a continuous-time stochastic process $(X_t)_$ is ''predictable'' if $X$, considered as a mapping from $\Omega \times \mathbb_$, 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

References

{{Reflist

Stochastic processes