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Predictable process

## Summary

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.[clarification needed]

## Mathematical definition

### Discrete-time process

Given a filtered probability space ${\displaystyle (\Omega ,{\mathcal {F}},({\mathcal {F}}_{n})_{n\in \mathbb {N} },\mathbb {P} )}$ , then a stochastic process ${\displaystyle (X_{n})_{n\in \mathbb {N} }}$  is predictable if ${\displaystyle X_{n+1}}$  is measurable with respect to the σ-algebra ${\displaystyle {\mathcal {F}}_{n}}$  for each n.[1]

### Continuous-time process

Given a filtered probability space ${\displaystyle (\Omega ,{\mathcal {F}},({\mathcal {F}}_{t})_{t\geq 0},\mathbb {P} )}$ , then a continuous-time stochastic process ${\displaystyle (X_{t})_{t\geq 0}}$  is predictable if ${\displaystyle X}$ , considered as a mapping from ${\displaystyle \Omega \times \mathbb {R} _{+}}$ , is measurable with respect to the σ-algebra generated by all left-continuous adapted processes.[2] This σ-algebra is also called the predictable σ-algebra.

## Examples

• Every deterministic process is a predictable process.[citation needed]
• Every continuous-time adapted process that is left continuous is obviously a predictable process.[citation needed]