Let $B_t$ be a standard brownian motion (starting at 0) and let $$U_t:=U_0+at,\quad L_t=-L_0-bt,\quad U_0, L_0, a, b>0$$ be two linear barriers, say. Is there a closed form of the probability $$\Bbb P(L_t\le B_t\le U_t,\, \forall t\ge 0)$$
By closed form I mean "as analytical as possible". But of course semi-closed forms like special functions or numerical integrals are also accepted. The bottomline is that it should be a low-computational cost expression and can be easily programmed, as opposed to costly numerical algorithms like MC etc.
Single linear barrier cases can be solved via change of measure and optional stopping etc. What about the double linear barrier case?
For what it's worth, this is equivalent to the probability of a Brownian Bridge staying within double flat barriers.
UPDATE - Given up the hopes of finding anything simpler than an infinite series representation. So infinite series (desirably with nice convergence properties) are all welcome.