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

摘要

Let ({mu _{t}^{(i)}}_{tge 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)}}_{tge 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 (tge 0). We end up with a possible application in mathematical finance.