Law (stochastic processes)

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.


Let (Ω, "F", P) be a probability space, "T" some index set, and ("S", Σ) a measurable space. Let "X" : "T" × Ω → "S" be a stochastic process (so the map

:X_{t} : Omega o S : omega mapsto X (t, omega)

is a ("F", Σ)-measurable function for each "t" ∈ "T"). Let "S""T" denote the collection of all functions from "T" into "S" (see remark below). The process "X" induces a function Φ"X" : Ω → "S""T", where

:left( Phi_{X} (omega) ight) (t) := X_{t} (omega).

The law of the process "X" is then defined to be the pushforward measure

:mathcal{L}_{X} := left( Phi_{X} ight)_{*} ( mathbf{P} )

on "S""T".

(Cautious readers may wonder for a moment if "S""T" really is a set. Abstractly, a function "T" → "S" is a certain type of subset of the Cartesian product "T" × "S", so the collection of all functions "T" → "S" is just a collection of certain elements of the power set of "T" × "S", and so is definitely a set.)


* The law of standard 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.)

ee also

* Finite-dimensional distribution

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