Law (stochastic processes)
Encyclopedia
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.
, T some index set, and (S, Σ) a measurable space. Let X : T × Ω → S be a stochastic process (so the map
is a (F, Σ)-measurable function
for each t ∈ T). Let ST denote the collection of all functions from T into S. The process X (by way of currying
) induces a function ΦX : Ω → ST, where
The law of the process X is then defined to be the pushforward measure
on ST.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the law of a stochastic process
Stochastic process
In probability theory, a stochastic process , or sometimes random process, is the counterpart to a deterministic process...
is the measure
Measure (mathematics)
In mathematical analysis, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset. In this sense, a measure is a generalization of the concepts of length, area, and volume...
that the process induces on the collection of functions
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
from the index set
Index set
In mathematics, the elements of a set A may be indexed or labeled by means of a set J that is on that account called an index set...
into the state space. The law encodes a lot of information about the process; in the case of a random walk
Random walk
A random walk, sometimes denoted RW, is a mathematical formalisation of a trajectory that consists of taking successive random steps. For example, the path traced by a molecule as it travels in a liquid or a gas, the search path of a foraging animal, the price of a fluctuating stock and the...
, for example, the law is the probability distribution
Probability measure
In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as countable additivity...
of the possible trajectories of the walk.
Definition
Let (Ω, F, P) be a probability spaceProbability space
In probability theory, a probability space or a probability triple is a mathematical construct that models a real-world process consisting of states that occur randomly. A probability space is constructed with a specific kind of situation or experiment in mind...
, T some index set, and (S, Σ) a measurable space. Let X : T × Ω → S be a stochastic process (so the map
is a (F, Σ)-measurable function
Measurable function
In mathematics, particularly in measure theory, measurable functions are structure-preserving functions between measurable spaces; as such, they form a natural context for the theory of integration...
for each t ∈ T). Let ST denote the collection of all functions from T into S. The process X (by way of currying
Currying
In mathematics and computer science, currying is the technique of transforming a function that takes multiple arguments in such a way that it can be called as a chain of functions each with a single argument...
) induces a function ΦX : Ω → ST, where
The law of the process X is then defined to be the pushforward measure
Pushforward measure
In measure theory, a pushforward measure is obtained by transferring a measure from one measurable space to another using a measurable function.-Definition:...
on ST.
Example
- The law of standard Brownian motionBrownian motionBrownian motion or pedesis is the presumably random drifting of particles suspended in a fluid or the mathematical model used to describe such random movements, which is often called a particle theory.The mathematical model of Brownian motion has several real-world applications...
is classical Wiener measure. (Indeed, many authors define Brownian motion to be a sample continuous processSample continuous processIn mathematics, a sample-continuous process is a stochastic process whose sample paths are almost surely continuous functions.-Definition:Let be a probability space. Let X : I × Ω → S be a stochastic process, where the index set I and state space S...
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.)
See also
- Finite-dimensional distributionFinite-dimensional distributionIn 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 onto a finite-dimensional vector space .-Finite-dimensional distributions of a measure:Let be a measure space...
- stochastic processStochastic processIn probability theory, a stochastic process , or sometimes random process, is the counterpart to a deterministic process...