2021/04/05 by Yuu Hariya
Computer Science · Economics, Econometrics and Finance · Mathematics · #Algebraic Geometry and Number Theory #Brownian motion #Calculus (dental) #Classical mechanics #Combinatorics #Complex Systems and Time Series Analysis #Computer science #Diffusion process #Extension (predicate logic) #Financial Risk and Volatility Modeling #Geometric Brownian motion #Geometry #Girsanov theorem #Identity (music) #Law #Mathematical economics #Mathematical physics #Mathematics #Mathematics and Applications #Motion (physics) #Path (computing) #Physics #Political science #Polynomial and algebraic computation #Pure mathematics #Quadratic equation #Quadratic variation #Stochastic processes and financial applications #math.PR #msc:60G30 #msc:60J55 #msc:60J65
paper · pdf · open access · doi:10.1016/j.spa.2022.01.005
published in Stochastic Processes and their Applications 146, 311-334 (Elsevier BV) · 28 pages
arxiv created 2021/04/05 · openalex publication_date 2022/01/13 · arxiv updated 2022/01/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 e^Bt, t≥ 0. Bougerol's celebrated identity (1983) asserts that, if β =\ β (t)\ t≥ 0 is another Brownian motion independent of B, then β (At) is identical in law with \sinh Bt for every fixed t>0. 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.