Lévy processes in finance: a remedy to the non-stationarity of continuous martingales

Marc Yor () and Boris Leblanc ()
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Marc Yor: Université Paris VI, Laboratoire de Probabilités, Tour 56, 4 place Jussieu, F-75252 Paris Cedex 05, France Manuscript
Boris Leblanc: Ingenierie Options G.I.E./Groupe BNP, Université Paris VII, 13, rue La Fayette, F-75009 Paris, France

Finance and Stochastics, 1998, vol. 2, issue 4, 399-408

Abstract: In this note, we prove that under some minor conditions on $\sigma$, if a martingale $X_t = \int_0^t \sigma_u dW_u $ satisfies, for every given pair $u \geq 0, \, \xi \geq 0$, $X_{u+\xi}-X_u{\mathop{=}^{\mathrm{(law)}}} X_{\xi},$ then necessarily, $|\sigma_u|$ is a constant and X is a constant multiple of a Brownian motion, thus providing a partial analogue of Lévy's characterisation of Brownian motion. In the introduction we explain why this theorem is a reason for considering Lévy processes in finance.

Keywords: Levy processes; martingales with stationary increments; forward-start-options (search for similar items in EconPapers)
JEL-codes: G13 (search for similar items in EconPapers)
Date: 1998-07-27
Note: received: May 1997; final version received: November 1997
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