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 (\Omega, \mathcal{F}, \mathbb{P}) be a probability space. Let X : I \times \Omega \to \mathbb{X} be a stochastic process (so the map

X_{i} : \Omega \to \mathbb{X} : \omega \mapsto X (i, \omega)

is a measurable function for each i \in I). Let \mathbb{X}^{I} denote the collection of all functions from I into \mathbb{X} (see remark below.) The process X induces a function \Phi_{X} : \Omega \to \mathbb{X}^{I}, where

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

The law of X is then defined to be the push forward measure

\mathcal{L}_{X} := \left( \Phi_{X} \right)_{*} ( \mathbb{P} ) \mathrm{\, on \,} \mathbb{X}^{I}.

(Cautious readers may wonder for a moment if \mathbb{X}^{I} really is a set. Abstractly, a function I \to \mathbb{X} is a certain type of subset of the Cartesian product I \times \mathbb{X}, so the collection of all functions I \to \mathbb{X} is just a collection of certain elements of the power set of I \times \mathbb{X}, 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.)

[edit] See also

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