Overview - 2023.2 English

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2023.2 English

A Brownian bridge is a continuous-time stochastic process \(B(t)\) whose probability distribution is the conditional probability distribution of a Wiener process \(W(t)\) subject to the condition (when standardized) that \(W(T) = 0\). More precisely:

\[B_{t} := (W_{t} | W_{T} = 0), t \in [0, T]\]

Generally, a Brownian bridge can be defined as:

Suppose \(\{W_{t}\}_{t \in [0,T]}\) is an 1-dimensional Brownian motion, \(a, b \in \mathbb{R}\), then the process

\[B^{a,b}_{t} = a\dfrac{T-t}{T} + b\dfrac{t}{T} + (W_{t} - \dfrac{t}{T}W_{T}), t \in [0, T]\]

is a Brownian bridge from \(a\) to \(b\).

It satisfies

\[B^{a,b}_{t} \sim \mathcal N (a + \dfrac{t}{T}(b-a), t - \dfrac{t^2}{T})\]

where \(\mathcal{N}\) is normal probability distribution.