Extensions of Bougerol’s identity in law and the associated anticipative path transformations

Abstract

Let B={Bt}t≥0 be a one-dimensional standard Brownian motion and denote by At,t≥0, the quadratic variation of the geometric Brownian motion eBt,t≥0. Bougerol’s celebrated identity in law (1983) asserts that, if β={β(t)}t≥0 is another Brownian motion independent of B, then, for every fixed t>0, β(At) is identical in law with sinhBt. In this paper, we extend Bougerol’s identity to an identity in law for processes up to time t, which exhibits a certain invariance of the law of Brownian motion. The extension is described in terms of anticipative transforms of B involving At as an anticipating factor. A Girsanov-type formula for those transforms is shown. An extension of a variant of Bougerol’s identity is also presented.