Brownian bridge

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

A Brownian bridge is a continuous-time gaussian process B(t) whose probability distribution is the conditional probability distribution of a standard Wiener process W(t) (a mathematical model of Brownian motion) 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:

$B_t:=(W_t\mid W_T=0),t\in [0,T]$

The expected value of the bridge at any t in the interval [0,T] is zero, with variance $\frac{t(T-t)}{T}$, 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 $\min (s,t)-\frac{st}{T}$, 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 $t\ge 0$, $W(t)$ is normally distributed with expected value $0$ and variance $t$, and the increments are stationary and independent), then

$B(t)=W(t)-\frac{t}{T}W(T)$

is a Brownian bridge for $t\in [0,T]$. It is independent of $W(T)$

Conversely, if $B(t)$ is a Brownian bridge for $t\in [0,1]$ and $Z$ is a standard normal random variable independent of $B$, then the process

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

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

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

Another representation of the Brownian bridge based on the Brownian motion is, for $t\in [0,T]$

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

Conversely, for $t\in [0,\mathrm{\infty }]$

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

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

$B_t=\sum _{k=1}^{\mathrm{\infty }}{Z}_k\frac{\sqrt{2T}\sin (k\pi t/T)}{k\pi }$

where $Z_1,Z_2,\dots$ 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.

Let $K=\underset{t\in [0,1]}{\sup }|B(t)|$, then the cumulative distribution function of $K$ is given by$$\mathrm{Pr}(K\le x)=1-2\sum _{k=1}^{\mathrm{\infty }}(-1{)}^{k-1}{e}^{-2k^2{x}^2}=\frac{\sqrt{2\pi }}{x}\sum _{k=1}^{\mathrm{\infty }}{e}^{-(2k-1{)}^2{\pi }^2/(8x^2)}.$$

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 [0,T]. (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,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 W(t₁) = a and W(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 variance

$\frac{(t_2-t)(t-t_1)}{t_2-t_1},$

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

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

References

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