Brownian bridge
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A Brownian bridge is a continuous-time
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 appea ...
''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 phenomenon i ...
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 ...
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 is ...
''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: : B_t := (W_t\mid W_T=0),\;t \in ,T The expected value of the bridge at any ''t'' in the interval ,''T''is zero, with variance \textstyle\frac, 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 ...
of ''B''(''s'') and ''B''(''t'') is \min(s,t)-\frac, 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 with expected value 0 and variance ''t'', and the increments are stationary and independent), then : B(t) = W(t) - \frac W(T)\, 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 Normal(s) or The Normal(s) may refer to: Film and television * ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson * ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie * ''Norma ...
random variable independent of ''B'', then the process : W(t) = B(t) + tZ\, is a Wiener process for ''t'' ∈  , 1 More generally, a Wiener process ''W''(''t'') for ''t'' ∈  , ''T''can be decomposed into : W(t) = \sqrtB\left(\frac\right) + \frac Z. Another representation of the Brownian bridge based on the Brownian motion is, for ''t'' ∈  , ''T'' : B(t) = \frac W\left(\frac\right). Conversely, for ''t'' ∈  , ∞ : W(t) = \frac B\left(\frac\right). The Brownian bridge may also be represented as a Fourier series with stochastic coefficients, as : B_t = \sum_^\infty Z_k \frac where Z_1, Z_2, \ldots 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 us ...
standard normal random variables (see the Karhunen–Loève theorem). A Brownian bridge is the result of
Donsker's theorem In probability theory, Donsker's theorem (also known as Donsker's invariance principle, or the functional central limit theorem), named after Monroe D. Donsker, is a functional extension of the central limit theorem. Let X_1, X_2, X_3, \ldots be ...
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 wit ...
in the area of
statistical inference Statistical inference is the process of using data analysis to infer properties of an underlying probability distribution, distribution of probability.Upton, G., Cook, I. (2008) ''Oxford Dictionary of Statistics'', OUP. . Inferential statistical ...
.


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 Normal(s) or The Normal(s) may refer to: Film and television * ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson * ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie * ''Norma ...
, 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 ''arithme ...
:a + \frac(b-a) 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 numbers ...
:\frac, 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 ...
between ''B''(''s'') and ''B''(''t''), with ''s'' < ''t'' is :\frac.


References

* * {{Authority control Wiener process Empirical process