We consider a geometric Lévy market with asset price S t = S 0 exp(X t ), where X is a
general Lévy process on (Ω, F, P), and interest rate equal to zero. As it is well known,
except for the cases that X is a Brownian motion or a Poisson process, the market is
incomplete. Therefore, if the market is arbitrage-free, there are many equivalent mar-
tingale measures and the problem arises to choose an appropriate martingale measure
for pricing contingent claimes.
One way is to choose the equivalent martingale measure Q ∗ which minimizes the
relative entropie to P, if it exists. Another choice is the famous Esscher martingale
measure Q E , if it exists.
The main objective of the present talk is to discuss a simple and rigorous approach
for proving the fact that the entropie minimal martingale measure Q ∗ and the Esscher
martingale measure Q E actually coincide: Q ∗ = Q E . Our method consists of a suit-
able approximation of the physical probability measure P by Lévy preserving probaility
measures P n.
The problem was treated in several earlier papers but more heuristally or in a so-