Brownian bridge

Brownian motion, pinned at both ends. This uses a Brownian bridge.

A Brownian bridge is a continuous-time stochastic process B(t) whose probability distribution is the conditional probability distribution of a Wiener process W(t) (a mathematical model of Brownian motion) given the condition that B(1) = 0. More precisely:

$B_t := (W_t\mid W_1=0),\;t \in [0,1]$

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 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), then

$B(t) = W(t) - t W(1)\,$

is a Brownian bridge for t ∈ [0, 1]. It is independent of W(1)

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

$W(t) = B(t) + tZ\,$

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

$W(t) = B\left(\frac{t}{T}\right) + \frac{t}{\sqrt{T}} Z.$

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

$B(t) = (1-t) W\left(\frac{t}{1-t}\right).$

Conversely, for t ∈ [0, ∞]

$W(t) = (1+t) B\left(\frac{t}{1+t}\right).$

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

$B_t = \sum_{k=1}^\infty Z_k \frac{\sqrt{2} \sin(k \pi t)}{k \pi}$

where $Z_1, Z_2, \ldots$ are independent identically distributed standard normal random variables (see the Karhunen–Loève theorem).

A Brownian bridge is the result of Donsker's theorem in the area of empirical processes. It is also used in the Kolmogorov–Smirnov test 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(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

$a + \frac{t-t_1}{t_2-t_1}(b-a)$

and the covariance between B(s) and B(t), with s < t is

$\frac{(t_2-t)(s-t_1)}{t_2-t_1}.$