On absolute continuity and singularity of multidimensional diffusions

Abstract

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.