# Probability that a $d$-dimensional Brownian bridge is greater than a given parameter

Let $$(W_t)_{t\in[0,T]}$$ be a Brownian bridge such that $$W_0=a$$ and $$W_T=b$$, the probability that $$\forall t\in[0,T],W_t\geqslant x$$ given the parameter $$x\leqslant\min(a,b)$$ is well known : $$\mathbb{P}(\forall t\in[0,T],W_t\geqslant x)=1-e^{\frac{2(x-a)(b-x)}{T}}$$ (see https://mathoverflow.net/questions/269096/probability-of-general-brownian-or-non-bridge-to-be-higher-than-given-paramete for a proof). My question is : Is there a generalization of this result for $$d$$-dimensional Brownian bridges ? That is, if $$(W_t)_{t\in[0,T]}$$ is a $$d$$-dimensional Brownian bridge such that $$W_0=a\in\mathbb{R}^d$$ and $$W_T=b\in\mathbb{R}^d$$, what is the probability that for all $$t\in [0,T],\|W_t\|\geqslant x$$ where $$x>0$$, for a convenient norm $$\|\cdot\|$$ whether it is $$\|\cdot\|_2$$, $$\|\cdot\|_{\infty}$$ or any norm that makes it possible to compute/approximate.

In addition, if we know enough about $$\mathbb{P}(\forall t\in[0,T],\|W_t\|\geqslant x)$$, what about $$\mathbb{E}[\mu\left(\{t\in[0,T],\|W_t\|\geqslant x\}\right)]$$ which is the average time spent above $$x$$ ?

• The requirement that $W_0 = a,W_T = b$ makes these problems not very tractable. In this case, $||W_t||^2 = (\sum B^{a_i,b_i}_t)^2$, where $B^{a_i,b_i}$ are independent Brownian bridges from $a_i$ to $b_i$. So $||W||^2$ has a rather complicated generalized $\xi^2$ distribution, which means that there unlikely to be nice (or any) expressions for these probabilities. Commented May 1, 2021 at 11:44
• However, if you instead look at $||W_t|| = a$, $||W_t|| = b$, this becomes much more manageable, as you are working with Bessel($d$) process, which is quite well understood. I would expect a more or less closed-form expression at least for the second question with $a=b=0$. Commented May 1, 2021 at 11:47
• Thanks for the answer, I can't find any article tackling this subject. In particular, the fact that there is no reflection principle for Bessel process (as far as I know) makes it more difficult to generalize the proof on Wiener process. Do you have any references ? Commented May 2, 2021 at 14:11
• Unfortunately, I don't. Try searching for "Bessel bridges" or "pinned Bessel processes". Commented May 2, 2021 at 14:20
• Ok I'll do it, thanks anyway. Commented May 2, 2021 at 14:45