Uniqueness of the embedding continuous convolution semigroup of a Gaussian probability measure on the affine group and an application in mathematical finance

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.