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. 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.

[edit] 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 \to S : \omega \mapsto X (t, \omega)

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

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

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

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

on ST.

(Cautious readers may wonder for a moment if ST 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.)

[edit] 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.)

[edit] See also