Université Paris 6
Pierre et Marie Curie | Université Paris 7
Denis Diderot | |

CNRS U.M.R. 7599
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``Probabilités et Modèles Aléatoires''
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**Auteur(s): **

- E. EBERLEIN
- J. JACOD
**S. RAIBLE**

**Code(s) de Classification MSC:**

- 60G51 Processes with independent increments
- 91B28 Finance, portfolios, investment
- 60G44 Martingales with continuous parameter
- 60J75 Jump processes

**Résumé:** We study the term structure models which are driven by a Lévy
process, from the point of view of arbitrage and completeness. Exactly
as for the Heath--Jarrow--Morton model, which fits into
our class of models, we observe that the conditions on the
coefficients for having no arbitrage
opportunity are rather stringent. For the completeness problem, the
results are quite
surprising: namely the model is complete if the driving Lévy process
is $1$--dimensional, provided the coefficient are non--random, or they
satisfy a very mild non--degeneracy assumption if they are random. On
the other hand, an example suggests that the model is no longer
complete when the Lévy process is genuinely multidimensional. This
is in deep contrast with the completeness of stock prices models,
where typically we have completeness if the number of stock prices is
bigger than or equal to the dimension of the driving Brownian motion
in the continuous case, while completeness usually fails when the
driving process is discontinuous.

**Mots Clés:** *Lévy processes ; representation of martingales ; term structure model ;
Heath-Jarrow-Morton ; arbitrage*

**Date:** 2003-10-16

**Prépublication numéro:** *PMA-854*