Brownian bridge explained

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\mid} W _{T=0), t} \in [ 0 ,T]

The expected value of the bridge at any t in the interval [0,''T''] is zero, with variance

style (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 min (s,t) - (st)/(T) , or s(T - t)/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 $t\; \backslash geq\; 0$, $W(t)$ is normally distributed with expected value $0$ and variance $t$, and the increments are stationary and independent), then

B(t) = W(t) - (t)/(T) W(T)

is a Brownian bridge for $t\; \backslash in\; [0,\; T]$. It is independent of $W(T)$^[1]

Conversely, if $B(t)$ is a Brownian bridge and $Z$ is a standard normal random variable independent of $B$, then the process

W(t) = B(t) + tZ

is a Wiener process for $t\; \backslash in\; [0,\; 1]$. More generally, a Wiener process $W(t)$ for $t\; \backslash in\; [0,\; T]$ can be decomposed into

W(t) = \sqrt{T}B\left( (t)/(T) \right) + (t)/(\sqrt{T) } Z.

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

B(t) = (T - t)/(W\left( \sqrt T) (t)/(T - t) \right).

Conversely, for $t\; \backslash in\; [0,\; \backslash infty]$

W(t) = (T + t)/(B\left( T) (Tt)/(T + t) \right).

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

B _{t} = \sum ^{inf ty} _{k =1} Z _{k} (\sqrt{ 2 T)/(\sin(k) \pi t / T)}{k \pi}

where

Z _{1,} Z _{2,} \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 = \sup _{t\in[0,1]} |B(t)| , then the cumulative distribution function of $K$ is given by^[2] $$\backslash operatorname(K\backslash leq\; x)=1-2\backslash sum\_^\backslash infty\; (-1)^\; e^=\backslash frac\backslash sum\_^\backslash infty\; e^.$$

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 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

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

a + (t - t _{1})/(t _{2-t} _{1}) (b - a)

and variance

((t _{2-t)(t-t} _{1)})/(t _{2-t} _{1}) ,

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

((t _{2-t)(s-t} _{1)})/(t _{2-t} _{1}) .

References

• Book: Glasserman, Paul . Monte Carlo Methods in Financial Engineering . 0-387-00451-3 . Springer-Verlag . New York . 2004.
• Book: Revuz, Daniel . Yor, Marc . Continuous Martingales and Brownian Motion . 2nd . 3-540-57622-3 . Springer-Verlag . New York . 1999.

Notes and References

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 . 10.18637/jss.v008.i18 . free.