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.
