Robust replication in H-self-similar Gaussian market models under uncertainty

We consider the robust hedging problem in the framework of model uncertainty,where the log-returns of the stock price are Gaussian andH-self-similar withH∈(1/2,1).Theseassumptions lead to two natural but mutually exclusive hypotheses, both being self-contained to fixthe probabilistic model for the stock price. Namely, the investor may assume that either the marketis efficient, that is the stock price process is a continuous semimartingale, or that the centred log-returns have stationary distributions. We show that to be able to super-hedge a European contingentclaim with a convex payoff robustly, the investor must assume that the markets are efficient. Ifit turns out that the stationarity hypothesis is true, then the investor can actually super-hedge theoption and thereby receive some net profit.