Uniqueness of the Embedding Continuous Convolution Semigroup of a Gaussian Probability Measure on the Affine Group and an Application in Mathematical Finance

Let {μ(i)t}t≥0 ( i=1,2 ) be continuous convolution semigroups (c.c.s.) of probability measures on Aff(1) (the affine group on the real line). Suppose that μ(1)1=μ(2)1 . Assume furthermore that {μ(1)t}t≥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 μ(1)t=μ(2)t for all t≥0 . We end up with a possible application in mathematical finance.