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.^{[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] 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.^{[citation needed]}

See also

• Adapted process
• Martingale

References