I would like to calculate $$E(1_{\{a \leq m+ vX \leq b\}}1_{\{c \leq n+ w(\rho X+ \sqrt{1-\rho^2} Y) \leq d\}})$$
where $ -1 \leq \rho \leq 1$ , $v>0, w >0, a<b, c<d$ and $X,Y$ normally distributed centered with variance 1, independent.
This expectation can be written as a double integral of the form $$\int_{m_1}^{m_2}[\Phi(m_3-\alpha_3x)-\Phi(m_4-\alpha_4x)] f_X(x)dx$$
where $m_1,m_2,m_3,m_4,\alpha_3$ and $\alpha_4$ are functions of the initial parameters. $f_X$ is the density function of a normal(reduced and centered) variable.
WHat is next then ? Is there a closed-form formula for that ?