On Absolute Continuity and Singularity of Multidimensional Diffusions

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.

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