Financial Markets with Uncertain Volatility

 Abstract.  Even in simple models in which thevolatility is only known to stay in two bounds, it is quite hard to price andhedge derivatives which are not Markovian.  The main reason for thisdifficulty emanates from the fact that the probability measures are singular toeach other.  In this talk we will prove a martingale representation theoremfor this market.  This result provides a complete answer to the questionsof hedging and pricing.  The main tools are the theory of nonlinearG-expectations as developed by Peng, the quasi-sure sto chastic artini and thesecond order backward stochastic differential equations.

 This is jointwork with Nizar Touzi from Ecole Polytechnique and Jianfeng Zhang fromUniversity of Southern California.