Let $B_t$ be a standard BM, is there an analytical or semi-analytical formula for the CDF of $(B_{t_1}, B_{t_2},\cdots, B_{t_n})$? Here $t_1<\cdots<t_n$, and by CDF I mean the probability $$\Bbb P(B_{t_1}<x_1,\cdots,B_{t_n}<x_n).$$
Low dimensionality cases seem simple e.g. $n=2$ or $3$ because we can somehow "invert" the covariance between the few $B_{t_i}$s and get an integration region under standard multivariate normal distribution. We can get the region right because dimensionality is low and our intuition works. But this no longer works for, say, $n\ge 4$.
Can anyone help?