Given two correlated Brownian Motions $X, Y$ starting at $0$ with correlation $\rho$ and respective drifts and variances $\mu_X, \mu_Y, \sigma_X^2, \sigma_Y^2$, define moving-maxima $X^*_t$ and $Y^*_t$ as $$X^*_t=\max_{0\le s\le t}X_s.$$ Find the probability $$\mathbb{P}(X^*_t<a, Y^*_t>b)$$ for $a,b,t>0$.
Does anyone know how to solve this or have suggestions?
see the work in Exact asymptotics of component-wise extrema of two-dimensional Brownian motion
at least for the tail behaviour.