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:

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 ${\frac {t(T-t)}{T}}$ , 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 $\min(s,t)-{\frac {s\,t}{T}}$ , or ${\frac {s(T-t)}{T}}$ if $s<t$ . The increments in a Brownian bridge are not independent.

Decomposition by zero-crossings

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

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

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

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 ${\textstyle t\in [0,T]}$

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

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

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

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

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

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

which can be conditionally sampled as

\tau _{+}={\frac {1}{1+\sin ^{2}\left({\frac {\pi }{2}}U_{1}\right)}}\in \left({\frac {1}{2}},1\right)

\tau _{-}={\frac {U_{2}^{2}\tau _{+}}{2\tau _{+}+U_{2}^{2}-1}}\in \left(0,{\frac {1}{2}}\right)

where $U_{1},U_{2}$ are uniformly distributed random variables over (0,1).

Intuitive remarks

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

and variance

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

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

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

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Decomposition by zero-crossings

Intuitive remarks

General case

References

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