Consider general Brownian bridge W(0)=0; W(T) = a. (Here ["general"][1] means:  $W(T)\ne 0$).

What is the probability   W(t) >= b, for all $ t \in [0, T] $  ? 

Is there close simple formula in terms of a, b  ,  T  ?

Can the formula be the same for any martingale type stochastic process (e.g. random walk), not only brownian motion, or it somehow depends on the details of stochastic process  ?
If there is such dependence how the questioned probability will change if we consider the distribition W(t) to have more and more "heavy tails" ? 


(I'm sure that questions are well-known for experts, but it is somehow difficult to google the asnwer, so let me ask here). 

Remark:
If we consider  somewhat informally related question - probability for W first hit a, before hitting (-b), (i.e. W(some T)=a and W(t < T) > b)  there is simple formula P = b/(a+b), which holds true for any martingale stochastic process. 
The questions are somewhat different, but still resemble each other and so  the simplicity of the answer P=b/(a+b) makes me hope that the answer to my question  might be simple and closed form. 
 


  [1]: https://en.wikipedia.org/wiki/Brownian_bridge#General_case