Predictable process

In stochastic analysis, a part of the mathematical theory of probability, a predictable process is a stochastic process whose value is knowable^{[clarification needed]} at a prior time. The predictable processes form the smallest class^{[clarification needed]} 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]

Examples

• Every deterministic process is a predictable process.^{[citation needed]}
• Every continuous-time process that is left continuous is a predictable process.^{[citation needed]}

See also

• Adapted process
• Martingale

References

The first reference is outdated, the link does not work.

1. van Zanten, Harry (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.