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Constructing a Brownian Bridge Between Fixed Endpoints

Article Quant Q&A · Author: Jack
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Summary

The document describes a Brownian bridge from a fixed starting value to a fixed ending value over a time interval. It begins with the standard representation on an interval starting at zero: a linear interpolation between the endpoints plus a Brownian motion adjusted by its terminal value. This construction makes the endpoint constraints explicit and provides a way to characterize the process through Brownian motion.

A reply gives a related construction starting at a general initial time, combining Brownian motion with terms that force the process to equal the prescribed values at both ends. Setting the initial time to zero recovers a familiar bridge representation. The original question asks about the distribution over a subinterval, but the response presents a process construction rather than stating the requested distribution explicitly. The material is therefore useful for defining and building bridges, while leaving the subinterval distribution and any application-specific interpretation to be derived.

Key ideas

  • A Brownian bridge can be formed from Brownian motion by adding a correction that enforces fixed endpoint values.
  • The mean path between the endpoints is represented by linear interpolation.
  • A construction can be written for an interval beginning at a general initial time.
  • The response does not explicitly provide the requested subinterval distribution.

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# What is the distribution of Brownian Bridge over a given time interval?


# What is the distribution of Brownian Bridge over a given time interval?












I know from Karatzas & Shreve (1991) that a Brownian Bridge $B(t)$ from $a$ to $b$ on time interval $[0,T]$ satisfies:

$$B(t)=a(1-t/T) + b*t/T + [W(t) - W(T)*t/T]$$

where $W(t)$ is a standard one-dimensional Brownian motion.

By the above equation we can get its distribution.

My question is what's the distribution of the Brownian Bridge $B(t)$ from $a$ to $b$ on time interval $[T_1, T_2]$?

Any idea or reference?

## Answer by KACEFMA. (score 2)

https://quant.stackexchange.com/a/25499

A Brownian bridge can be built simply from a forced process. For example, if we define the process $Z$ by

$$ Z_{t}=\left(\dfrac{T-t}{T-t_0}\right)\left(a-W_{t_0}\right)+\left(\dfrac{t-t_0}{T-t_0}\right)\left(b-W_{T}\right)+W_{t}\;\;\;; t\in[t_0,T] $$

Where $W$ is a standard Brownian motion.

The process $Z$ is a Brownian bridge satisfying the following conditions: $Z_{t_0}=a$ and $Z_T=b$, in particular, if we put $t_0=0$, Then the process $Z$ becomes $$ Z_{t}=W_{t}+a-\frac{t}{T}\left(W_{T}-b+a\right) $$

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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