In stochastic analysis, a part of the mathematical theory of probability

Probability

Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...

, a predictable process is a stochastic process

Stochastic process

In probability theory, a stochastic process , or sometimes random process, is the counterpart to a deterministic 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

Adapted process

In the study of stochastic processes, an adapted process is one that cannot "see into the future". An informal interpretation is that X is adapted if and only if, for every realisation and every n, Xn is known at time n...

 left continuous processes.

Discrete-time process

Given a filtered probability space , then a stochastic process is predictable if is measureable with respect to the σ-algebra  for each n.

Continuous-time process

Given a filtered probability space , then a continuous-time stochastic process

Continuous-time stochastic process

In probability theory and statistics, a continuous-time stochastic process, or a continuous-space-time stochastic process is a stochastic process for which the index variable takes a continuous set of values, as contrasted with a discrete-time process for which the index variable takes only...

  is predictable if is measureable with respect to the σ-algebra for each time t.

Examples

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