In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, 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 in a probability space, where the index of the family often has the interpretation of time. Sto ...
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 (with respect to the probability measure). In other words, the set of outcomes on which the event does not occur ha ...
continuous function
In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as '' discontinuities''. More preci ...
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 ...
. 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 Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
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 quantity". More precisely, if X is a set (mathematics), set, "almost all elements of X" means "all elements of X but those in a negligible set, negligible subset of X". The meaning o ...
''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 have many applications throughout pure mathematics an ...
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
*
{{Stochastic processes
Stochastic processes