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

## Author Info

• 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)

Registered author(s):

## 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.

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## Bibliographic Info

Article provided by Springer in its journal Finance and Stochastics.

Volume (Year): 2 (1998)
Issue (Month): 4 ()
Pages: 399-408

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