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Predictable process

Stochastic process

Stochastic 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 a predictable process.

References

References

  1. (November 8, 2004). "An Introduction to Stochastic Processes in Continuous Time".
  2. "Predictable processes: properties".
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