Brownian bridge

Brownian motion, pinned at both ends. This represents a Brownian bridge.

A Brownian bridge is a continuous-time stochastic 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 $\textstyle {\frac {t(T-t)}{T}}$, 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 $\min(s,t)-{\frac {s\,t}{T}}$, 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 ≥ 0, W(t) is normally distributed with expected value 0 and variance t, and the increments are stationary and independent), then

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

is a Brownian bridge for t ∈ [0, T]. It is independent of W(T)

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 ∈ [0, 1]. More generally, a Wiener process W(t) for t ∈ [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 t ∈ [0, T]

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

Conversely, for t ∈ [0, ∞]

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

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. Brownian bridges are also used in financial modeling to estimate the probability distribution of a stock price that cannot be assumed to follow a Geometric Brownian motion but there is only information about its price process for a discrete set of points (for example, option maturities).

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 B(t₁) = a and B(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

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

References

• 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.