Brownian Bridge Construction and Why Its Paths Use the Endpoint

Summary

The document introduces a Brownian bridge over a fixed interval as Brownian motion adjusted by a linear term involving its value at the interval endpoint. It gives the resulting zero mean and time-dependent variance, then asks how the bridge can be known before the endpoint is reached and how it should be simulated. The central issue is the distinction between defining a random path over a whole interval and observing that path sequentially in real time.

The questions point toward an important modeling caveat: the displayed construction depends on the terminal Brownian value, so constructing a bridge from a simulated Brownian path uses that endpoint when forming earlier bridge values. The document itself offers no answer or simulation procedure. It is therefore a conceptual prompt, not evidence that bridge values are unavailable in every simulation setting or a guide to a particular sampling algorithm.

Key ideas

• A Brownian bridge is defined by adjusting Brownian motion using its endpoint value.
• Its stated mean is zero, and its variance changes over the interval.
• The endpoint dependence raises a distinction between whole-path construction and sequential observation.
• The document poses simulation questions but does not provide their answers.

Tags

# Conceptual questions on Brownian Bridge


# Conceptual questions on Brownian Bridge












Brownian Bridge, denoted $B_t$ can be described as a Brownian motion pinned at zero and $T$, with the following mathematical definition:

$$B_t=W_t-\frac{t}{T}W_T$$

From the above, we can deduce that $\mathbb{E}[B_t]=0$ and $\mathbb{E}[B_t^2]=\frac{t(T-t)}{T}$.

I am unclear about the following conceptual points:

- For all $t\leq T$, how can we know the value of $W_T$? The value of $W_T$ is not known before we get to $T$

- Related to the point above, in a practical simulation, how would the values of $B_t$ for $t\leq T$ be computed? Would we first need to simulate $W_t$ all the way up to $T$ and then construct the Bridge values "in hindsight" by looking up the values of $W_t$ for $t<T$ to compute the actual values of $B_t$ ?

- Combining the two points above, it sort of sounds like we cannot tell what $B_t$ is for any $t<T$, until we reach $T$: is this a correct conclusion?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.