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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 $x$ ?

Tuvasbien
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    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. – zhoraster May 01 '21 at 11:44
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    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 $a=b=0$. – zhoraster May 01 '21 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 ? – Tuvasbien May 02 '21 at 14:11
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    Unfortunately, I don't. Try searching for "Bessel bridges" or "pinned Bessel processes". – zhoraster May 02 '21 at 14:20
  • Ok I'll do it, thanks anyway. – Tuvasbien May 02 '21 at 14:45
  • I’m sorry you did not get a response for such a great question. You can also ask on math overflow, math tutors beta, statistics and even physics se. 500 reputation . is now gone. – Тyma Gaidash May 06 '21 at 00:03
  • It's fine, reputation exists to be spent. I'll post the question on math overflow, hopefully I'll get a nice answer, though I know it is a tough question. – Tuvasbien May 06 '21 at 00:14

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