In
mathematics, a sample-continuous process is a
stochastic process 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 functions.
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 consis ...
''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 po ...
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 mathem ...
''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">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 ...
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