Predictable process

In stochastic analysis, a part of the mathematical theory of probability, a predictable process is a stochastic process which the 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.

1 Mathematical definition
- 1.1 Discrete-time process
- 1.2 Continuous-time process
2 Examples
3 See also
4 References

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 $X_{n%2B1}$ is measureable 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 $X_{t}$ is measureable with respect to the σ-algebra $\mathcal{F}_{t^-}$ for each time t.^[2]

Examples

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

References

^ van Zanten, Harry (November 8, 2004). "An Introduction to Stochastic Processes in Continuous Time" (pdf). http://www.cs.vu.nl/~rmeester/onderwijs/stochastic_processes/sp_new.pdf. Retrieved October 14, 2011.
^ "Predictable processes: properties" (pdf). http://www.math.ku.dk/~jesper/teaching/b108/slides38.pdf. Retrieved October 15, 2011.