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Brownian Bridge

A Brownian bridge from a to b over the time interval [0,T] is a Brownian motion conditioned to start at a and end at b. If W_t is a standard Brownian motion, then a Brownian bridge with diffusion coefficient D can be constructed as

X_t=a+t/T(b-a)+sqrt(2D)(W_t-t/TW_T).
(1)

It has expectation value and covariance

E[X_t]=a+t/T(b-a)
(2)
cov(X_s,X_t)=2D(min(s,t)-(st)/T).
(3)

In particular, X_0=a and X_T=b with probability one, and

var(X_t)=2D(t(T-t))/T.
(4)

Writing p_t(x,y) for the transition density of the unconditioned Brownian motion, the density of the bridge at an intermediate time is

p^(a,b,T)(x,t)=(p_t(a,x)p_(T-t)(x,b))/(p_T(a,b)).
(5)

This factorization is a special case of the Markov bridge formula. The same conditioned process satisfies the time-inhomogeneous stochastic differential equation

dX_t=(b-X_t)/(T-t)dt+sqrt(2D)dW_t.
(6)

This equation holds for 0<=t<T.

See also

Doob h-Transform, Markov Bridge, Wiener Process

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References

Fitzsimmons, P. J.; Pitman, J.; and Yor, M. "Markovian Bridges: Construction, Palm Interpretation, and Splicing." In Seminar on Stochastic Processes, 1992. Boston, MA: Birkhäuser, pp. 101-134, 1993. https://doi.org/10.1007/978-1-4612-0339-1_5.Karatzas, I. and Shreve, S. E. Brownian Motion and Stochastic Calculus, 2nd ed. New York: Springer-Verlag, 1991.

Cite this as:

Weisstein, Eric W. "Brownian Bridge." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/BrownianBridge.html

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