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 and H-self-similar with H∈(1/2,1). These assumptions lead to two natural but mutually exclusive hypotheses, both being self-contained to fix the probabilistic model for the stock price. Namely, the investor may assume that either the market is 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 contingent claim with a convex payoff robustly, the investor must assume that the markets are efficient. If it turns out that the stationarity hypothesis is true, then the investor can actually super-hedge the option and thereby receive some net profit.