关于多维扩散的绝对连续性和奇性
On Absolute Continuity and Singularity of Multidimensional Diffusions
摘要：
考虑多维可能爆炸扩散的两个定律\(P\)和\(Q\)，它们分别具有共同的扩散系数\(\mathfrak{a}\)和漂移系数\(\mathfrak{b}\)和\(\mathfrak{b} + \mathfrak{a} \mathfrak{c}\)，以及具有扩散系数\(\langle \mathfrak{c},\mathfrak{a}\mathfrak{c}\rangle^{-1}\mathfrak{a}\)和漂移系数\(\langle \mathfrak{c}, \mathfrak{a}\mathfrak{c}\rangle^{-1}\mathfrak{b}\)的辅助扩散的定律\(P^\circ\)。我们证明了\(P \ll Q\)当且仅当辅助扩散\(P^\circ\)几乎必然爆炸，而\(P\perp Q\)当且仅当辅助扩散\(P^\circ\)几乎必然不爆炸。作为应用，我们推导出绝对连续性和奇异性的卡斯明斯基型积分检验、时变布朗运动爆炸的积分检验，并讨论了在数学金融中的应用。
英文摘要：
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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