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Brownian bridge

From Wikipedia, the free encyclopedia
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:

The expected value of the bridge at any t in the interval [0,T] is zero, with variance , 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 , or s(T − t)/T if s < t. The increments in a Brownian bridge are not independent.

Relation to other stochastic processes

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If is a standard Wiener process (i.e., for , is normally distributed with expected value and variance , and the increments are stationary and independent), then

is a Brownian bridge for . It is independent of [1]

Conversely, if is a Brownian bridge for and is a standard normal random variable independent of , then the process

is a Wiener process for . More generally, a Wiener process for can be decomposed into

Another representation of the Brownian bridge based on the Brownian motion is, for

Conversely, for

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

where 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 , then the cumulative distribution function of is given by[2]

Intuitive remarks

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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

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For the general case when W(t1) = a and W(t2) = b, the distribution of B at time t ∈ (t1t2) is normal, with mean

and variance

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

References

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  1. ^ Aspects of Brownian motion, Springer, 2008, R. Mansuy, M. Yor page 2
  2. ^ Marsaglia G, Tsang WW, Wang J (2003). "Evaluating Kolmogorov's Distribution". Journal of Statistical Software. 8 (18): 1–4. doi:10.18637/jss.v008.i18.
  • Glasserman, Paul (2004). Monte Carlo Methods in Financial Engineering. New York: Springer-Verlag. ISBN 0-387-00451-3.
  • Revuz, Daniel; Yor, Marc (1999). Continuous Martingales and Brownian Motion (2nd ed.). New York: Springer-Verlag. ISBN 3-540-57622-3.