Multivariate normal distribution I have 4 r.v $(X,Y,W,Z)$ distributed as a multivariate normal. 
Mean and variance-covariance matrix are known.
Is it possible to calculate, for example 
$\mathrm{Prob} (aX+bY < k, W >0,Z >0, W-Z >0)$, where $k$ is a scalar?
If not, which additional information do I need?
 A: In general multivariate normal probabilities are non-trivial and not usually solvable algebraically. However, on your specific question we can get somewhere.
Note that if $(X,Y,W,Z)$ is multivariate normal then $(k-aX+bY,W,Z,W-Z)$ will also be mutivariate normal (if degenerate in this case), so the problem reduces to one of finding whether a multivariate normal vector is all positive.


*

*If the means of the new set of variates are zero this is an example of finding an orthant probability, a classical problem on which some results are available.
The book by Genz and Bretz (2009), Computation of Multivariate Normal and t Probabilities lists explicit formulas for up to trivariate normal, 
$P_2= \frac14+\frac{1}{2\pi}\sin^{-1}(\rho_{12})$
$P_3= \frac18+\frac{1}{4\pi}\{\sin^{-1}(\rho_{12})+\sin^{-1}(\rho_{23})+\sin^{-1}(\rho_{13}) \}$
and for even dimension gives a dimension-halving formula. As it happens, the relevant formulas are in the sample pages available on the book's web-page
But in your specific, case, we can in fact simplify further and directly apply the trivariate formula there (see the next section). 

*In the more general case of nonzero means. Note that if $Z>0$ and $W-Z>0$, then $W>0$ is automatic. Consequently, let our vector be $(k-aX+bY,Z,W-Z)$, compute its mean and covariance matrix; let $(T,U,V)$ be mean-corrected versions of the above vector. This is discussed in the trivariate section of the book that's also in the above linked extract.

*The paper by Genz 1992 ("Revised Computation of Multivariate normal probabilities," J.Comp. Graph. Stat, 1, pp. 141-149; working paper here) discusses a number of approximations for the general case.

*The R package mvtnorm includes algorithms for calculation of multivariate normal probabilities.
For high dimensional probabilities if I didn't have premade routines to call, I'd probably be leaning toward using various Monte Carlo methods (which are widely used for these problems). 
