Uniqueness of the embedding continuous convolution semigroup of a Gaussian probability measure on the affine group and an application in mathematical finance |
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Authors: | Daniel Neuenschwander |
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Institution: | 1. Faculté des HEC, Institut de Sciences Actuarielles, Université de Lausanne, 1015, Lausanne, Switzerland 2. Institut für mathematische Statistik und Versicherungslehre, Universit?t Bern, 3012, Bern, Switzerland
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Abstract: | Let $\{\mu _{t}^{(i)}\}_{t\ge 0}$ ( $i=1,2$ ) be continuous convolution semigroups (c.c.s.) of probability measures on $\mathbf{Aff(1)}$ (the affine group on the real line). Suppose that $\mu _{1}^{(1)}=\mu _{1}^{(2)}$ . Assume furthermore that $\{\mu _{t}^{(1)}\}_{t\ge 0}$ is a Gaussian c.c.s. (in the sense that its generating distribution is a sum of a primitive distribution and a second-order differential operator). Then $\mu _{t}^{(1)}=\mu _{t}^{(2)}$ for all $t\ge 0$ . We end up with a possible application in mathematical finance. |
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