Law (stochastic processes)
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In mathematics, the law of a stochastic process is the measure that the process induces on the collection of functions from the index set into the state space.
[edit] Definition
Let 
be a probability space. Let 
be a stochastic process (so the map
is a measurable function for each 
). Let 
denote the collection of all functions from I into 
(see remark below.) The process X induces a function 
, where
The law of X is then defined to be the pushforward measure
(Cautious readers may wonder for a moment if really is a set. Abstractly, a function 
is a certain type of subset of the Cartesian product 
, so the collection of all functions is just a collection of certain elements of the power set of , and so is definitely a set.)
[edit] Example
- The law of Brownian motion is classical Wiener measure. (Indeed, many authors define Brownian motion to be a sample continuous process starting at the origin whose law is Wiener measure, and then proceed to derive the independence of increments and other properties from this definition; other authors prefer to work in the opposite direction.)
