Let $B_t$ be the standard Brownian motion and $L_t^a$ be the local time at level $a$. It is known that the joint-density of $(L_t^0,B_t)$ is

$$ P\left(B_t\in d y, L_t^0\in d v\right) = \frac{|y|+v}{\sqrt{2\pi t^3}}\exp\left(-\frac{(|y|+v)^2}{2 t}\right) 1_{[0,\infty)}(v)1_{(-\infty,\infty)}(y) d y d v, $$

see, e.g., p. 181 of Chung & Williams.

The problem is whether anyone knows the joint density of $(L_t^a,B_t)$ for $a\ne 0$?

Thank you very much for any hints.