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.
Given a filtered probability space ( Ω , F , ( F n ) n ∈ N , P ) , then a stochastic process ( X n ) n ∈ N is predictable if X n + 1 is measurable with respect to the σ-algebra F n for each n.
Given a filtered probability space ( Ω , F , ( F t ) t ≥ 0 , P ) , then a continuous-time stochastic process ( X t ) t ≥ 0 is predictable if X , considered as a mapping from Ω × R + , is measurable with respect to the σ-algebra generated by all left-continuous adapted processes.
Every deterministic process is a predictable process.Every continuous-time adapted process that is left continuous is a predictable process.