# 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/… ? – Quantuple Jun 18 '16 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. – OldSchool Jun 19 '16 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? – Mark Joshi Jun 21 '16 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 – Mark Joshi Jun 21 '16 at 3:25
• also it is a feature of the Brownian bridge that drift doesn't matter – Mark Joshi Jun 22 '16 at 7:34

$$P\textbf{(}min(x_0 \to _T) \leq \alpha\:\: | \:\: x_T\textbf{)} =\left\{\begin{matrix} e^{-2(\alpha -x_0)(\alpha -x_T)/\sigma ^{2}T} \:\:\:\:\:\:\:\:\ for\: \alpha \leq x_0\: and \: \alpha \leq x_T\ \\ \\ 1\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: Otherwise \end{matrix}\right.$$
• 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? – OldSchool Jul 2 '16 at 13:15