Conditional Brownian Barrier-Hitting Probability Given the Endpoint

Summary

The document asks for the probability that a Brownian motion with constant drift and volatility touches a lower barrier by a fixed horizon, conditional on its ending at a known value. It gives an unconditional barrier-touching formula as background, then reports a conditional result in piecewise form. When both the starting point and ending point lie above the barrier, the conditional probability is an exponential function of their distances from the barrier, volatility, and elapsed time. If either endpoint is at or below the barrier, the reported probability is one.

The result is attributed to a response that credits a pointer from another researcher. The document says the author had checked the unconditional formula against simulations, but it provides no corresponding derivation, conditional simulation, or detailed assumptions. The expression is presented for the stated drifted Brownian process and fixed endpoint; readers applying it should verify the parameter convention and applicability to their process, especially if using it for asset prices or barrier products.

Key ideas

• The target quantity is the chance of touching a lower barrier conditional on a known terminal value.
• When both endpoints are above the barrier, the reported probability decays exponentially with their distances from it.
• If either endpoint is at or below the barrier, the conditional touch probability is one.
• The result concerns a Brownian process with constant drift and volatility over a fixed horizon.
• The document supplies the formula but does not derive or independently validate the conditional expression.

Tags

# Probability of Brownian motion particle touching barrier given path starts at $X_0$ and ends at a known $X_t$


# Probability of Brownian motion particle touching barrier given path starts at $X_0$ and ends at a known $X_t$












I have been reading Su and Rieger's paper on barriers and from there have been able to work out the unconditional probability of the process $dXt = μ dt + σ dWt$ touching a down barrier $α$ to be

$\mathbb{P}(\min(x_0\rightarrow _T) ≤ α) = \Phi\left(\frac{α - μT}{σ \sqrt{T}}\right) + \exp\left(\frac{2μα}{σ^2}\right) \Phi\left(\frac{α + μT}{σ\sqrt{T}}\right)$

All well and good matching simulations nicely etc...

However, I am looking for a closed form solution for $\mathbb{P}(\min(x_0\rightarrow _T) ≤ α\, |\, x_T = X)$ (i.e both $x_0$ and $x_T$ are known.)

## Answer by OldSchool (score 0, accepted)

https://quant.stackexchange.com/a/27907

Thanks to Mark Joshi for pointing me to the answer in his comments above. Credit should go to him. For completeness here is the answer expressed in the nomenclature of the question.

$$ P\textbf{(}min(x_0 \to _T) \leq \alpha\:\: | \:\: x_T\textbf{)} =\left\{\begin{matrix} e^{-2(\alpha -x_0)(\alpha -x_T)/\sigma ^{2}T} \:\:\:\:\:\:\:\:\ for\: \alpha \leq x_0\: and \: \alpha \leq x_T\ \\ \\ 1\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: Otherwise \end{matrix}\right. $$

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.