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Brownian Bridge Barrier-Hit Probability from Endpoint and Extremum Laws

Article Quant Q&A · Author: OldSchool
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Summary

The document asks for the probability that a Brownian bridge, conditioned on its terminal value, reaches an upper barrier before a lower barrier within a fixed horizon. It starts from the familiar upper-before-lower hitting probability for arithmetic Brownian motion, then explains why conditioning on both endpoints calls for a bridge-specific calculation.

The proposed approach expresses the desired event as the probability of staying above the lower barrier minus the probability of remaining between both barriers, conditional on the endpoint. It then gives an attempted expression using Gaussian transition densities and an infinite reflection-series sum for paths constrained between barriers. This is presented as a calculation attempt based on hitting-time results, not a verified derivation; notation and normalization details may need checking. No numerical examples or empirical validation are supplied.

Key ideas

  • The target probability conditions on both the starting and ending values of the process.
  • The event can be decomposed using probabilities of avoiding the lower barrier and staying between both barriers.
  • Gaussian densities and a reflection-series expression are used in the proposed calculation.
  • The formula is explicitly an attempt and is not accompanied by validation or examples.

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# What is the probability that a Brownian Bridge hits an upper barrier $U$ before a lower barrier $L$?


# What is the probability that a Brownian Bridge hits an upper barrier $U$ before a lower barrier $L$?












The probability that an arithmetic Brownian motion process $dt = \mu dt + \sigma dW$ hits an upper Barrier $U$ before it hits a lower barrier $L$ is given by

$$ \mathbb{P}(\tau_U\leq \tau_L) = \frac{\text{Y}(x_0)-\text{Y}(L)}{\text{Y}(U)-\text{Y}(L)} $$ where $$ \text{Y}(x) = exp(\frac{-2\mu x}{\sigma^2}) $$

But what is $\mathbb{P}(\tau_U\leq T \,\cap\, \tau_U\leq\tau_L)$ if both $x_0$ and $x_T$ are known?

i.e. the probability the process hits $U$ before $L$ whilst in between the end points of a Brownian bridge.

## Answer by M. Jeunesse (score 1)

https://quant.stackexchange.com/a/28176

#### Idea

Let $B$ be a standard brownian motion starting from $x_0=0$, $m_T = \inf_{u\leq T}B_u$ and $M_T =\sup_{u\leq T}B_u$.

Let's define if it exists for $A\in\sigma(B_u,u\leq T)$, $\mathbb{P}(A | B_T=x_T)\stackrel{\rm def}{=}\lim_{\varepsilon\to 0}\mathbb{P}(A|B_T\in(x_T-\varepsilon,x_T+\varepsilon))$

$$\begin{split} \mathbb{P}(\tau_U\leq T \cap \tau_U\leq \tau_L)= & \mathbb{P}(m_T>L;M_T\geq U |B_T =x_T)\\ = &\mathbb{P}(m_T>L|B_T=x_T)-\mathbb{P}(L<m_T<M_T<U|B_T=x_T) \end{split}$$

#### Computations attempt

Then, this is side computations based from results about hitting times from Chapter 3 of Mathematical Methods for Financial Markets of Monique Jeanblanc, Marc Yor and Marc Chesney.

so by denoting $p_T(y)=\frac{1}{\sqrt{2\pi t}}e^{-\frac{y^2}{2t}}$

I have: $$\begin{split} P\stackrel{\rm def}{=} & \mathbb{P}(\tau_U\leq T \cap \tau_U\leq \tau_L)\\ =& \frac{p_T(-2L+x_T)}{p_T(-x_T)} - \frac{\sum_{n=-\infty}^{\infty}p_T(x_T+2n(U-L))-p_T(2U-x_T+2n(U-L))}{p_T(x_T)} \end{split}$$

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.