Law (stochastic processes)
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. The law encodes a lot of information about the process; in the case of a random walk, for example, the law is the probability distribution of the possible trajectories of the walk.Definition
Let be a probability space, T some index set, and a measurable space. Let X : T × Ω → S be a stochastic process. Let ST denote the collection of all functions from T into S. The process X induces a function ΦX : Ω → ST, where
The law of the process X is then defined to be the pushforward measure
on ST.Example
