In
stochastic analysis, a part of the mathematical theory of
probability
Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, ...
, a predictable process is a
stochastic process
In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that ap ...
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
Discrete-time process
Given a
filtered probability space
Filtration is a physical separation process that separates solid matter and fluid from a mixture using a ''filter medium'' that has a complex structure through which only the fluid can pass. Solid particles that cannot pass through the filte ...
, then a stochastic process
is ''predictable'' if
is
measurable
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many simila ...
with respect to the
σ-algebra for each ''n''.
Continuous-time process
Given a filtered probability space
, then a
continuous-time stochastic process is ''predictable'' if
, considered as a mapping from
, is measurable with respect to the σ-algebra generated by all left-continuous adapted processes.
This
σ-algebra
In mathematical analysis and in probability theory, a σ-algebra (also σ-field) on a set ''X'' is a collection Σ of subsets of ''X'' that includes the empty subset, is closed under complement, and is closed under countable unions and countabl ...
is also called the predictable σ-algebra.
Examples
* Every
deterministic process is a predictable process.
* Every continuous-time adapted process that is
left continuous
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in val ...
is obviously a predictable process.
See also
*
Adapted process In the study of stochastic processes, an adapted process (also referred to as a non-anticipating or non-anticipative process) is one that cannot "see into the future". An informal interpretation is that ''X'' is adapted if and only if, for every rea ...
*
Martingale
References
{{Reflist
Stochastic processes