Abstract
Hamza-Klebaner posed the problem of constructing martingales with one-dimensional Brownian marginals that differ from Brownian motion, so called fake Brownian motions. Besides its theoretical appeal, the problem represents the quintessential version of the ubiquitous fitting problem in mathematical finance where the task is to construct martingales that satisfy marginal constraints imposed by market data. Non-continuous solutions to this challenge were given by Madan-Yor, Hamza-Klebaner, Hobson and Fan-Hamza-Klebaner, whereas continuous (but non-Markovian) fake Brownian motions were constructed by Oleszkiewicz, Albin, Baker-Donati-Yor, Hobson and Jourdain-Zhou. In contrast it is known from Gyöngy, Dupire, and ultimately Lowther that Brownian motion is the unique continuous strong Markov martingale with Brownian marginals. We took this as a challenge to construct examples of a "barely fake" Brownian motion, that is, continuous Markov martingales with one-dimensional Brownian marginals that miss out only on the strong Markov property.