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.

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


Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Stochastic processes and boundary value problems — In mathematics, some boundary value problems can be solved using the methods of stochastic analysis. Perhaps the most celebrated example is Shizuo Kakutani s 1944 solution of the Dirichlet problem for the Laplace operator using Brownian motion.… …   Wikipedia

  • Infinitesimal generator (stochastic processes) — In mathematics mdash; specifically, in stochastic analysis mdash; the infinitesimal generator of a stochastic process is a partial differential operator that encodes a great deal of information about the process. The generator is used in… …   Wikipedia

  • Law (disambiguation) — Law is a set of norms, which can be seen both in a sociological or in a philosophical or semantic sense.Law or laws may also refer to:Government*Rule of law, the principle that restricts governmental authority *Items associated with the practice… …   Wikipedia

  • Power law — A power law is any polynomial relationship that exhibits the property of scale invariance. The most common power laws relate two variables and have the form:f(x) = ax^k! +o(x^k),where a and k are constants, and o(x^k) is of x. Here, k is… …   Wikipedia

  • Kolmogorov's zero-one law — In probability theory, Kolmogorov s zero one law, named in honor of Andrey Nikolaevich Kolmogorov, specifies that a certain type of event, called a tail event , will either almost surely happen or almost surely not happen; that is, the… …   Wikipedia

  • Gibrat's law — Gibrat s law, sometimes called Gibrat s rule of proportionate growth is a rule defined by Robert Gibrat (1904 1980) stating that the size of a firm and its growth rate are independent. Gibrat s law is also applied to cities size and growth rate… …   Wikipedia

  • List of mathematics articles (L) — NOTOC L L (complexity) L BFGS L² cohomology L function L game L notation L system L theory L Analyse des Infiniment Petits pour l Intelligence des Lignes Courbes L Hôpital s rule L(R) La Géométrie Labeled graph Labelled enumeration theorem Lack… …   Wikipedia

  • Finite-dimensional distribution — In mathematics, finite dimensional distributions are a tool in the study of measures and stochastic processes. A lot of information can be gained by studying the projection of a measure (or process) onto a finite dimensional vector space (or… …   Wikipedia

  • probability theory — Math., Statistics. the theory of analyzing and making statements concerning the probability of the occurrence of uncertain events. Cf. probability (def. 4). [1830 40] * * * Branch of mathematics that deals with analysis of random events.… …   Universalium

  • Convergence of random variables — In probability theory, there exist several different notions of convergence of random variables. The convergence of sequences of random variables to some limit random variable is an important concept in probability theory, and its applications to …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”