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:

${\displaystyle B_{t}:=(W_{t}\mid W_{T}=0),\;t\in [0,T]}$

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

If ${\textstyle W(t)}$ is a standard Wiener process (i.e., for ${\textstyle t\geq 0}$, ${\textstyle W(t)}$ is normally distributed with expected value ${\textstyle 0}$ and variance ${\textstyle t}$, and the increments are stationary and independent), then

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

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

Conversely, if ${\textstyle B(t)}$ is a Brownian bridge for ${\textstyle t\in [0,1]}$ and ${\textstyle Z}$ is a standard normal random variable independent of ${\textstyle B}$, then the process

${\displaystyle W(t)=B(t)+tZ\,}$

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

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

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

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

Conversely, for ${\textstyle t\in [0,\infty ]}$

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

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

${\displaystyle B_{t}=\sum _{k=1}^{\infty }Z_{k}{\frac {{\sqrt {2T}}\sin(k\pi t/T)}{k\pi }}}$

where ${\displaystyle Z_{1},Z_{2},\ldots }$ 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 ${\displaystyle K=\sup _{t\in [0,1]}|B(t)|}$, then the cumulative distribution function of ${\textstyle K}$ is given by^[2]$${\displaystyle \operatorname {Pr} (K\leq x)=1-2\sum _{k=1}^{\infty }(-1)^{k-1}e^{-2k^{2}x^{2}}={\frac {\sqrt {2\pi }}{x}}\sum _{k=1}^{\infty }e^{-(2k-1)^{2}\pi ^{2}/(8x^{2})}.}$$

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).

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

${\displaystyle a+{\frac {t-t_{1}}{t_{2}-t_{1}}}(b-a)}$

and variance

${\displaystyle {\frac {(t_{2}-t)(t-t_{1})}{t_{2}-t_{1}}},}$

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

${\displaystyle {\frac {(t_{2}-t)(s-t_{1})}{t_{2}-t_{1}}}.}$

• 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.