# 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 , 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]}

## Examples

- Every deterministic process is a predictable process.
- Every continuous-time adapted process that is left continuous is a predictable process.

## See also

## References

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

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