论文标题
多维扩散的绝对连续性和奇异性
On Absolute Continuity and Singularity of Multidimensional Diffusions
论文作者
论文摘要
考虑多维的两个定律\（p \）和\（q \），可能具有爆炸性扩散，并具有常见的扩散系数\（\ Mathfrak {a} \）和漂移系数\（\ Mathfrak {b}分别具有扩散系数\（\ langle \ mathfrak {c}，\ Mathfrak {a} \ Mathfrak {c} \ rangle^{c} \ rangle^{ - 1} { - 1} \ Mathfrak {a} {a} \ lang coftic coftic coftic \ lang coftic \ lang（ \ Mathfrak {C}，\ Mathfrak {A} \ Mathfrak {C} \ rangle^{ - 1} \ Mathfrak {B} \）。当辅助扩散\（p^\ crocr \）几乎肯定地爆炸时，我们显示\（p \ ll q \）时，并且仅当辅助扩散\（p \ perp q \）爆炸时，当时\（p \ perp q \）当辅助扩散\（p^\ circt \）几乎不爆炸。作为应用，我们为绝对连续性和奇异性提供了Khasminskii-Type积分测试，这是对时间变化的布朗运动爆炸的积分测试，我们讨论了对数学金融的应用。
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.
