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 (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.)

Example

* 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