Brownian bridge

 ( Wiener Process ) 

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:

$B\_t\; :=\; (W\_t\backslash mid\; W\_T=0),\backslash ;t\; \backslash in\; 0,T$

The expected value of the bridge at any $t$ in the interval $0,T$ is zero, with variance $\backslash frac\{t(T-t)\}\{T\}$, 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 $\backslash min(s,t)-\backslash frac\{s\backslash ,t\}\{T\}$, or $\backslash frac\{s(T-t)\}\{T\}$ if . The increments in a Brownian bridge are not independent.

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Relation to other stochastic processes If $W(t)$ is a standard Wiener process (i.e., for $t\; \backslash geq\; 0$, $W(t)$ is normally distributed with expected value $0$ and variance $t$, and the increments are stationary and independent), then

$B(t)\; =\; W(t)\; -\; \backslash frac\{t\}\{T\}\; W(T)\backslash ,$

is a Brownian bridge for $t\; \backslash in\; 0,$. It is independent of $W(T)$Aspects of Brownian motion, Springer, 2008, R. Mansuy, M. Yor page 2

Conversely, if $B(t)$ is a Brownian bridge for $t\; \backslash in\; 0,$ and $Z$ is a standard normal random variable independent of $B$, then the process

$W(t)\; =\; B(t)\; +\; tZ\backslash ,$

is a Wiener process for $t\; \backslash in\; 0,$. More generally, a Wiener process $W(t)$ for $t\; \backslash in\; 0,$ can be decomposed into

$W(t)\; =\; \backslash sqrt\{T\}B\backslash left(\backslash frac\{t\}\{T\}\backslash right)\; +\; \backslash frac\{t\}\{\backslash sqrt\{T\}\}\; Z.$

Another representation of the Brownian bridge based on the Brownian motion is, for $t\; \backslash in\; 0,$

$B(t)\; =\; \backslash frac\{T-t\}\{\backslash sqrt\; T\}\; W\backslash left(\backslash frac\{t\}\{T-t\}\backslash right).$

Conversely, for $t\; \backslash in\; 0,$

$W(t)\; =\; \backslash frac\{T+t\}\{T\}\; B\backslash left(\backslash frac\{Tt\}\{T+t\}\backslash right).$

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

$B\_t\; =\; \backslash sum\_\{k=1\}^\backslash infty\; Z\_k\; \backslash frac\{\backslash sqrt\{2\; T\}\; \backslash sin(k\; \backslash pi\; t\; /\; T)\}\{k\; \backslash pi\}$

where $Z\_1,\; Z\_2,\; \backslash ldots$ 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 $K=\backslash sup\_\{t\backslash in0,1\}|B(t)|$, for a Brownian bridge with $T\; =\; 1$; then the cumulative distribution function of $K$ is given by $$\backslash operatorname\{Pr\}(K\backslash leq\; x)=1-2\backslash sum\_\{k=1\}^\backslash infty\; (-1)^\{k-1\}\; e^\{-2k^2\; x^2\}=\backslash frac\{\backslash sqrt\{2\backslash pi\}\}\{x\}\backslash sum\_\{k=1\}^\backslash infty\; e^\{-(2k-1)^2\backslash pi^2/(8x^2)\}.$$

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Decomposition by zero-crossings The Brownian bridge can be "split" by finding the last zero $\backslash tau\_\{-\}$ before the midpoint, and the first zero $\backslash tau\_\{+\}$ after, forming a (scaled) bridge over $0,\backslash tau\_\{-\}$, an excursion over $\backslash tau\_\{-\},\backslash tau\_\{+\}$, and another bridge over $\backslash tau\_\{+\},1$. The joint pdf of $\backslash tau\_\{-\},\backslash tau\_\{+\}$ is given by

$\backslash rho\; \backslash left(\; \backslash tau\_\{-\},\backslash tau\_\{+\}\; \backslash right)\; =\; \backslash frac\{1\}\{2\; \backslash pi\; \backslash sqrt\{\backslash tau\_\{-\}\; (1-\backslash tau\_\{+\})\; (\backslash tau\_\{+\}-\backslash tau\_\{-\})^3\; \}\}$

which can be conditionally sampled as

$\backslash tau\_\{+\}\; =\; \backslash frac\{1\}\{1+\backslash sin^2\; \backslash left(\; \backslash frac\{\backslash pi\}\{2\}\; U\_1\; \backslash right)\}\; \backslash in\; \backslash left(\; \backslash frac\{1\}\{2\},\; 1\backslash right)$

$\backslash tau\_\{-\}\; =\; \backslash frac\{U\_2^2\; \backslash tau\_\{+\}\}\{2\backslash tau\_\{+\}\; +\; U\_2^2\; -1\}\; \backslash in\; \backslash left(\; 0,\backslash frac\{1\}\{2\}\; \backslash right)$

where $U\_1,U\_2$ are uniformly distributed random variables over (0,1).

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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( 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. The solution is to use a collection of T Brownian bridges, the first of which is required to go through the values W(0) and W(1), the second through W(1) and W(2) and so on until the Tth goes through W( T-1) and W( T).

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

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

and variance

$\backslash frac\{(t\_2-t)(t-t\_1)\}\{t\_2-t\_1\},$

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

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

• Paul Glasserman (2025). 9780387004518, Springer-Verlag. ISBN 9780387004518
• (1999). 9783540576228, Springer-Verlag. ISBN 9783540576228

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