A Brownian bridge is a continuous-time stochastic process

Stochastic process

In probability theory, a stochastic process , or sometimes random process, is the counterpart to a deterministic process...

 B(t) whose probability distribution

Probability distribution

In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....

 is the conditional probability distribution of a Wiener process

Wiener process

In mathematics, the Wiener process is a continuous-time stochastic process named in honor of Norbert Wiener. It is often called standard Brownian motion, after Robert Brown...

 W(t) (a mathematical model of Brownian motion

Brownian motion

Brownian motion or pedesis is the presumably random drifting of particles suspended in a fluid or the mathematical model used to describe such random movements, which is often called a particle theory.The mathematical model of Brownian motion has several real-world applications...

) given the condition that B(0) = B(1) = 0.

The expected value of the bridge is zero, with variance t(1 − t), implying that the most uncertainty is in the middle of the bridge, with zero uncertainty at the nodes. The covariance

Covariance

In probability theory and statistics, covariance is a measure of how much two variables change together. Variance is a special case of the covariance when the two variables are identical.- Definition :...

 of B(s) and B(t) is s(1 − 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

Lévy process

In probability theory, a Lévy process, named after the French mathematician Paul Lévy, is any continuous-time stochastic process that starts at 0, admits càdlàg modification and has "stationary independent increments" — this phrase will be explained below...

), then

is a Brownian bridge.

Conversely, if B(t) is a Brownian bridge and Z is a standard normal random variable, then the process

is a Wiener process for t ∈ [0, 1]. More generally, a Wiener process W(t) for t ∈ [0, T] can be decomposed into

Another representation of the Brownian bridge based on the Brownian motion is, for t ∈ [0, 1]

Conversely, for t ∈ [0, ∞]

The Brownian bridge may also be represented as a Fourier series with stochastic coefficients, as

where are independent identically distributed standard normal random variables (see the Karhunen–Loève theorem).

A Brownian bridge is the result of Donsker's theorem

Donsker's theorem

In probability theory, Donsker's theorem, named after M. D. Donsker, identifies a certain stochastic process as a limit of empirical processes. It is sometimes called the functional central limit theorem....

 in the area of empirical process

Empirical process

The study of empirical processes is a branch of mathematical statistics and a sub-area of probability theory. It is a generalization of the central limit theorem for empirical measures...

es. It is also used in the Kolmogorov–Smirnov test in the area of statistical inference

Statistical inference

In statistics, statistical inference is the process of drawing conclusions from data that are subject to random variation, for example, observational errors or sampling variation...

.

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(1) = 0, that is the process is "tied down" at t = 1 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 [0,1]. (In a slight generalization, one sometimes requires B(t₁) = a and B(t₂) = b where t₁, t₂, 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 [0,1], that is to interpolate between the already generated points W(0) and W(1). The solution is to use a Brownian bridge that is required to go through the values W(0) and W(1).

General case

For the general case when B(t₁) = a and B(t₂) = b, the distribution of B at time t ∈ (t₁, t₂) is normal, with mean

Expected value

In probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...

and variance

Variance

In probability theory and statistics, the variance is a measure of how far a set of numbers is spread out. It is one of several descriptors of a probability distribution, describing how far the numbers lie from the mean . In particular, the variance is one of the moments of a distribution...