Predictable process

In stochastic analysis, a predictable process is a stochastic process which the value is knowable at a prior time.

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 $X_{n+1}$ is measureable with respect to the σ-algebra ${\mathcal {F}}_{n}$ for each n.

Continuous time process

Given a filtered probability space $(\Omega ,{\mathcal {F}},({\mathcal {F}}_{t})_{t\geq 0},\mathbb {P} )$, then a 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.

Properties

• A continuous time process which is left continuous is always a predictable process.

See also

• Adapted process