Predictable process

From Infogalactic: the planetary knowledge core
Jump to: navigation, search

In stochastic analysis, a part of the mathematical theory of probability, a predictable process is a stochastic process whose value is knowable[clarification needed] at a prior time. The predictable processes form the smallest class[clarification needed] that is closed under taking limits of sequences and contains all adapted left-continuous processes[clarification needed].

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.[1]

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.[2]

Examples

See also

References

  1. Lua error in package.lua at line 80: module 'strict' not found.
  2. Lua error in package.lua at line 80: module 'strict' not found.


<templatestyles src="Asbox/styles.css"></templatestyles>