### Speaker

### Description

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-

phisticated way.