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.

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 measureable 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 stochastic process ${\displaystyle (X_{t})_{t\geq 0}}$  is predictable if ${\displaystyle X_{t}}$  is measureable with respect to the σ-algebra ${\displaystyle {\mathcal {F}}_{t^{-}}}$  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.

See also

• Adapted process

References

1. Harry van Zanten (November 8, 2004). "An Introduction to Stochastic Processes in Continuous Time" (pdf). Retrieved October 14, 2011.
2. "Predictable processes: properties" (pdf). Retrieved October 15, 2011.