In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a sample-continuous process is a
stochastic process
In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that ap ...
whose sample paths are
almost surely
In probability theory, an event is said to happen almost surely (sometimes abbreviated as a.s.) if it happens with probability 1 (or Lebesgue measure 1). In other words, the set of possible exceptions may be non-empty, but it has probability 0. ...
continuous function
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in val ...
s.
Definition
Let (Ω, Σ, P) be a
probability space
In probability theory, a probability space or a probability triple (\Omega, \mathcal, P) is a mathematical construct that provides a formal model of a random process or "experiment". For example, one can define a probability space which models t ...
. Let ''X'' : ''I'' × Ω → ''S'' be a stochastic process, where the
index set
In mathematics, an index set is a set whose members label (or index) members of another set. For instance, if the elements of a set may be ''indexed'' or ''labeled'' by means of the elements of a set , then is an index set. The indexing consists ...
''I'' and state space ''S'' are both
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
s. Then the process ''X'' is called sample-continuous (or almost surely continuous, or simply continuous) if the map ''X''(''ω'') : ''I'' → ''S'' is
continuous as a function of topological spaces for P-
almost all
In mathematics, the term "almost all" means "all but a negligible amount". More precisely, if X is a set, "almost all elements of X" means "all elements of X but those in a negligible subset of X". The meaning of "negligible" depends on the mathema ...
''ω'' in ''Ω''.
In many examples, the index set ''I'' is an interval of time,
, ''T''or
, +∞),_and_the_state_space_''S''_is_the_real_line_or_''n''-dimension.html" ;"title="real_line.html" ;"title=", +∞), and the state space ''S'' is the real line">, +∞), and the state space ''S'' is the real line or ''n''-dimension">real_line.html" ;"title=", +∞), and the state space ''S'' is the real line">, +∞), and the state space ''S'' is the real line or ''n''-dimensional Euclidean space R
''n''.
Examples
* Brownian motion (the Wiener process) on Euclidean space is sample-continuous.
* For "nice" parameters of the equations, solutions to
stochastic differential equation
A stochastic differential equation (SDE) is a differential equation in which one or more of the terms is a stochastic process, resulting in a solution which is also a stochastic process. SDEs are used to model various phenomena such as stock p ...
s are sample-continuous. See the existence and uniqueness theorem in the stochastic differential equations article for some sufficient conditions to ensure sample continuity.
* The process ''X'' : [0, +∞) × Ω → R that makes equiprobable jumps up or down every unit time according to
::
: is ''not'' sample-continuous. In fact, it is surely discontinuous.
Properties
* For sample-continuous processes, the finite-dimensional distributions determine the Law (stochastic processes), law, and vice versa.
See also
* Continuous stochastic process
References
* {{cite book
, author = Kloeden, Peter E.
, author2=Platen, Eckhard
, title = Numerical solution of stochastic differential equations
, series = Applications of Mathematics (New York) 23
, publisher = Springer-Verlag
, location = Berlin
, year = 1992
, pages = 38–39
, isbn = 3-540-54062-8
Stochastic processes