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 {μt (i)}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_1~((2)). Assume furthermore that {μ_t ~((1))}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_t ~((1))=μ_t~ ((2)) for all t≥ 0. We end up with a possible application in mathematical finance.