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]}

## Contents

## Mathematical definition

### Discrete-time process

Given a filtered probability space , then a stochastic process is *predictable* if is measurable with respect to the σ-algebra for each *n*.^{[1]}

### Continuous-time process

Given a filtered probability space , then a continuous-time stochastic process is *predictable* if , considered as a mapping from , 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.

## See also

## References

**^**van Zanten, Harry (November 8, 2004). "An Introduction to Stochastic Processes in Continuous Time" (PDF). Archived from the original (pdf) on April 6, 2012. Retrieved October 14, 2011.**^**"Predictable processes: properties" (PDF). Archived from the original (pdf) on March 31, 2012. Retrieved October 15, 2011.