Brownian bridge

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

A Brownian bridge is a continuous-time stochastic process B(t) whose probability distribution is the conditional probability distribution of a Wiener process W(t) (a mathematical model of Brownian motion) subject to the condition that W(T) = 0, so that the process is pinned at the origin 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 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 s(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)^[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 ∈ [0, T]. More generally, a Wiener process W(t) for t ∈ [0, T] can be decomposed into

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

Conversely, for t ∈ [0, ∞]

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

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

B_{t}=\sum _{k=T}^{\infty }Z_{k}{\frac {{\sqrt {2}}\sin(k\pi 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.

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

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

References

↑ Aspects of Brownian motion, Springer, 2008, R. Mansuy, M. Yor page 2

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.

Stochastic processes

Discrete time	Bernoulli process Branching process Chinese restaurant process Galton–Watson process Independent and identically distributed random variables Markov chain Moran process Random walk Loop-erased Self-avoiding

Continuous time	Bessel process Birth–death process Brownian motion Bridge Excursion Fractional Geometric Meander Cauchy process Contact process Continuous-time random walk Cox process Diffusion process Empirical process Feller process Fleming–Viot process Gamma process Hunt process Interacting particle systems Itô diffusion Itô process Jump diffusion Jump process Lévy process Local time Markov additive process McKean–Vlasov process Ornstein–Uhlenbeck process Poisson process Compound Non-homogeneous Point process Schramm–Loewner evolution Semimartingale Sigma-martingale Stable process Superprocess Telegraph process Variance gamma process Wiener process Wiener sausage

Both	Branching process Galves–Löcherbach model Gaussian process Hidden Markov model (HMM) Markov process Martingale Differences Local Sub- Super- Random dynamical system Regenerative process Renewal process Stochastic chains with memory of variable length White noise

Fields and other	Dirichlet process Gaussian random field Gibbs measure Hopfield model Ising model Potts model Boolean network Markov random field Percolation Pitman–Yor process Point process Cox Poisson Random field Random graph

Time series models	Autoregressive conditional heteroskedasticity (ARCH) model Autoregressive integrated moving average (ARIMA) model Autoregressive (AR) model Autoregressive–moving-average (ARMA) model Generalized autoregressive conditional heteroskedasticity (GARCH) model Moving-average (MA) model

Financial models	Black–Derman–Toy Black–Karasinski Black–Scholes Chen Constant elasticity of variance (CEV) Cox–Ingersoll–Ross (CIR) Garman–Kohlhagen Heath–Jarrow–Morton (HJM) Heston Ho–Lee Hull–White LIBOR market Rendleman–Bartter SABR volatility Vašíček Wilkie

Actuarial models	Bühlmann Cramér–Lundberg Risk process Sparre–Anderson

Queueing models	Bulk Fluid Generalized queueing network M/G/1 M/M/1 M/M/c

Properties	Càdlàg paths Continuous Continuous paths Ergodic Exchangeable Feller-continuous Gauss–Markov Markov Mixing Piecewise deterministic Predictable Progressively measurable Self-similar Stationary Time-reversible

Limit theorems	Central limit theorem Donsker's theorem Doob's martingale convergence theorems Ergodic theorem Fisher–Tippett–Gnedenko theorem Large deviation principle Law of large numbers (weak/strong) Law of the iterated logarithm Maximal ergodic theorem Sanov's theorem

Inequalities	Burkholder–Davis–Gundy Doob's martingale Kunita–Watanabe

Tools	Cameron–Martin formula Convergence of random variables Doléans-Dade exponential Doob decomposition theorem Doob–Meyer decomposition theorem Doob's optional stopping theorem Dynkin's formula Feynman–Kac formula Filtration Girsanov theorem Infinitesimal generator Itô integral Itô's lemma Kolmogorov continuity theorem Kolmogorov extension theorem Lévy–Prokhorov metric Malliavin calculus Martingale representation theorem Optional stopping theorem Prokhorov's theorem Quadratic variation Reflection principle Skorokhod integral Skorokhod's representation theorem Skorokhod space Snell envelope Stochastic differential equation Tanaka Stopping time Stratonovich integral Uniform integrability Usual hypotheses Wiener space Classical Abstract

Disciplines	Actuarial mathematics Econometrics Ergodic theory Extreme value theory (EVT) Large deviations theory Mathematical finance Mathematical statistics Probability theory Queueing theory Renewal theory Ruin theory Statistics Stochastic analysis Time series analysis Machine learning

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