Predictable process

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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.

Mathematical definition#edit

Discrete-time process#edit

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#edit

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#edit

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

References#edit

^ van Zanten, Harry (November 8, 2004). "An Introduction to Stochastic Processes in Continuous Time" (PDF). Archived from the original (pdf) on April 6, 2012. Retrieved October 14, 2011.
^ "Predictable processes: properties" (PDF). Archived from the original (pdf) on March 31, 2012. Retrieved October 15, 2011.

Categories Stochastic processes

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