Brownian bridge
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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 ≥ 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. 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).
A Brownian bridge is useful in several situations. First, 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). Furthermore, a Brownian bridge is the result of Donsker's theorem in the area of empirical processes.
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. For the general case when [W(T_1)=a] and [W(T_2)=b], the distribution of [W] at time [T\in (T_1,T_2)] is normal, with mean
- [a + \frac(b-a)]
- [\frac.]
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