  Consider two laws \(P\) and \(Q\) of multidimensional possibly explosive
diffusions with common diffusion coefficient \(\mathfrak{a}\) and drift
coefficients \(\mathfrak{b}\) and \(\mathfrak{b} + \mathfrak{a} \mathfrak{c}\),
respectively, and the law \(P^\circ\) of an auxiliary diffusion with diffusion
coefficient \(\langle
\mathfrak{c},\mathfrak{a}\mathfrak{c}\rangle^{-1}\mathfrak{a}\) and drift
coefficient \(\langle \mathfrak{c},
\mathfrak{a}\mathfrak{c}\rangle^{-1}\mathfrak{b}\). We show that \(P \ll Q\) if
and only if the auxiliary diffusion \(P^\circ\) explodes almost surely and that
\(P\perp Q\) if and only if the auxiliary diffusion \(P^\circ\) almost surely
does not explode. As applications we derive a Khasminskii-type integral test
for absolute continuity and singularity, an integral test for explosion of
time-changed Brownian motion, and we discuss applications to mathematical
finance.
