A Brownian bridge is a continuous-time
stochastic process ''B''(''t'') whose
probability distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomeno ...
is the
conditional probability distribution
In probability theory and statistics, given two jointly distributed random variables X and Y, the conditional probability distribution of Y given X is the probability distribution of Y when X is known to be a particular value; in some cases the c ...
of a standard
Wiener process
In mathematics, the Wiener process is a real-valued continuous-time stochastic process named in honor of American mathematician Norbert Wiener for his investigations on the mathematical properties of the one-dimensional Brownian motion. It i ...
''W''(''t'') (a mathematical model of
Brownian motion
Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas).
This pattern of motion typically consists of random fluctuations in a particle's position insi ...
) subject to the condition (when standardized) that ''W''(''T'') = 0, so that the process is pinned to the same value at both ''t'' = 0 and ''t'' = ''T''. More precisely:
:
The expected value of the bridge at any ''t'' in the interval
,''T''is zero, with variance
, implying that the most uncertainty is in the middle of the bridge, with zero uncertainty at the nodes. The
covariance
In probability theory and statistics, covariance is a measure of the joint variability of two random variables. If the greater values of one variable mainly correspond with the greater values of the other variable, and the same holds for the le ...
of ''B''(''s'') and ''B''(''t'') is
, or ''s''(T − ''t'')/T if ''s'' < ''t''.
The increments in a Brownian bridge are not independent.
Relation to other stochastic processes
If ''W''(''t'') is a standard Wiener process (i.e., for ''t'' ≥ 0, ''W''(''t'') is
normally distributed
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The parameter \mu is ...
with expected value 0 and variance ''t'', and the
increments are stationary and independent), then
:
is a Brownian bridge for ''t'' ∈
, ''T'' It is independent of ''W''(''T'')
[Aspects of Brownian motion, Springer, 2008, R. Mansuy, M. Yor page 2]
Conversely, if ''B''(''t'') is a Brownian bridge and ''Z'' is a standard
normal random variable independent of ''B'', then the process
:
is a Wiener process for ''t'' ∈
, 1 More generally, a Wiener process ''W''(''t'') for ''t'' ∈
, ''T''can be decomposed into
:
Another representation of the Brownian bridge based on the Brownian motion is, for ''t'' ∈
, ''T''
:
Conversely, for ''t'' ∈
, ∞
:
The Brownian bridge may also be represented as a Fourier series with stochastic coefficients, as
:
where
are
independent identically distributed
In probability theory and statistics, a collection of random variables is independent and identically distributed if each random variable has the same probability distribution as the others and all are mutually independent. This property is usua ...
standard normal random variables (see the
Karhunen–Loève theorem).
A Brownian bridge is the result of
Donsker's theorem in the area of
empirical process
In probability theory, an empirical process is a stochastic process that describes the proportion of objects in a system in a given state.
For a process in a discrete state space a population continuous time Markov chain or Markov population model ...
es. It is also used in the
Kolmogorov–Smirnov test
In statistics, the Kolmogorov–Smirnov test (K–S test or KS test) is a nonparametric test of the equality of continuous (or discontinuous, see Section 2.2), one-dimensional probability distributions that can be used to compare a sample wi ...
in the area of
statistical inference.
Intuitive remarks
A standard Wiener process satisfies ''W''(0) = 0 and is therefore "tied down" to the origin, but other points are not restricted. In a Brownian bridge process on the other hand, not only is ''B''(0) = 0 but we also require that ''B''(''T'') = 0, that is the process is "tied down" at ''t'' = ''T'' as well. Just as a literal bridge is supported by pylons at both ends, a Brownian Bridge is required to satisfy conditions at both ends of the interval
,''T'' (In a slight generalization, one sometimes requires ''B''(''t''
1) = ''a'' and ''B''(''t''
2) = ''b'' where ''t''
1, ''t''
2, ''a'' and ''b'' are known constants.)
Suppose we have generated a number of points ''W''(0), ''W''(1), ''W''(2), ''W''(3), etc. of a Wiener process path by computer simulation. It is now desired to fill in additional points in the interval
,''T'' that is to interpolate between the already generated points ''W''(0) and ''W''(''T''). The solution is to use a Brownian bridge that is required to go through the values ''W''(0) and ''W''(''T'').
General case
For the general case when ''B''(''t''
1) = ''a'' and ''B''(''t''
2) = ''b'', the distribution of ''B'' at time ''t'' ∈ (''t''
1, ''t''
2) is
normal, with
mean
There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value ( magnitude and sign) of a given data set.
For a data set, the '' ari ...
:
and
variance
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of number ...
:
and the
covariance
In probability theory and statistics, covariance is a measure of the joint variability of two random variables. If the greater values of one variable mainly correspond with the greater values of the other variable, and the same holds for the le ...
between ''B''(''s'') and ''B''(''t''), with ''s'' < ''t'' is
:
References
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{{Authority control
Wiener process
Empirical process