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

## Author

Listed:
• 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)

## 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.

## Suggested Citation

• Marc Yor & Boris Leblanc, 1998. "Lévy processes in finance: a remedy to the non-stationarity of continuous martingales," Finance and Stochastics, Springer, vol. 2(4), pages 399-408.
• Handle: RePEc:spr:finsto:v:2:y:1998:i:4:p:399-408
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### Keywords

Levy processes; martingales with stationary increments; forward-start-options;

### JEL classification:

• G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing

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