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.

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" &ge; 0, "W"("t") is normally distributed with expected value 0 and variance "t", and the increments are stationary and independent), then "W"("t") − "t W"(1) is a Brownian bridge.

Conversely, if "B" is a Brownian bridge and "Z" is an independent standard Gaussian random variable, then the process $W(t)\; =\; B(t)\; +\; t\; Z$ is a Wiener process for "t" &isin; [0, "1"] .More generally, a Wiener process $W(t)$ for "t" &isin; [0, "T"] can be decomposed into

:$x(t)\; =\; Bleft(frac\{t\}\{T\}\; ight)\; +\; frac\{t\}\{sqrt\{T\; Z.$

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"("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 [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 "W"("t"₁) = "a" and "W"("t"₂) = "b", the distribution of "W" at time "t" &isin; ("t"₁, "t"₂) is normal, with mean

:$a\; +\; frac\{t-t\_1\}\{t\_2-t\_1\}(b-a)$

and variance

:$frac\{(t-t\_1)(t\_2-t)\}\{t\_2-t\_1\}.$

References

* Glasserman, Paul. "Monte Carlo Methods in Financial Engineering", ISBN 0-387-00451-3, Springer-Verlag New York, 2004