# 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.)

• Could you please use Latex to format your equations? It's very hard to understand what you want exactly. Does this help you: math.stackexchange.com/questions/412470/… ? Jun 18, 2016 at 0:06
• @Quantuple thanks for the link. I am looking into wether or not it can help me indirectly, but with my rather low experience I am having trouble getting from the infinite set of probability distributions of all Xts along the path to the distribution of the minimum in the path. Jun 19, 2016 at 18:00
• There's a well-known formula for $\mathbb{P}(\min(x_0\rightarrow _T) ≤ α\, x_T \geq x)$ can you not just differentiate it? Jun 21, 2016 at 3:21
• see S. Metwally, A. Atiya, Using Brownian bridge for fast simulation of jump-diffusion processes and barrier options, Journal of Derivatives, Fall 2002, 43--54, or Using Monte Carlo simulation and importance sampling to rapidly obtain jump-diffusion prices of continuous barrier options MS Joshi, T Leung - Available at SSRN 907386, 2005 Jun 21, 2016 at 3:25
• also it is a feature of the Brownian bridge that drift doesn't matter Jun 22, 2016 at 7:34

## 1 Answer

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.

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

• Having gone through this exercise I realize that what I really need is the probability that the Bronian bridge touches the down barrier $\alpha$ BEFORE it touches an up barrier $\beta$. I think I should post a further question rather than continuing here. Any objections? Jul 2, 2016 at 13:15