Brownian bridge



A Brownian bridge is a continuous-time stochastic process whose probability distribution is the conditional probability distribution of a Wiener process B(t) (a mathematical model of Brownian motion) given the condition that B(0) = B(1) = 0. Equivalently, 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 W(t) &minus; t W(1) is a Brownian bridge. The increments in a Brownian bridge are clearly not independent.

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).

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.

There is also a natural decomposition of a Brownian motion B(t) for t &isin; [0, n] into


 * $$B(t) = W_n(t) + \frac{t}{\sqrt{n}} Z$$

where Wn(t) is a Brownian bridge over time interval [0, n] and Z is a standard Gaussian random variable.

The expected value of the bridge is zero, with variance t(1 &minus; 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 &minus; t) if s < t.

For the general case when W(T1) = a and W(T2) = b, the distribution of W at time T &isin; (T1, T2) is normal, with mean


 * $$a + \frac{T-T_1}{T_2-T_1}(b-a)$$

and variance


 * $$\frac{(T-T_1)(T_2-T)}{T_2-T_1}.$$