Faking Brownian motion with continuous Markov martingales

Hamza and Klebaner (2007) [10] 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, this 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 and Yor (2002) [22], Hamza and Klebaner (2007) [10], Hobson (2016) [11] and Fan et al. (2015) [8], whereas continuous (but non-Markovian) fake Brownian motions were constructed by Oleszkiewicz (2008) [23], Albin (2008) [1], Baker et al. (2006) [4], Hobson (2013) [14], Jourdain and Zhou (2020) [16]. In contrast, it is known from Gyöngy (1986) [9], Dupire (1994) [7] and ultimately Lowther (2008) [17] and Lowther (2009) [20] that Brownian motion is the unique continuous strong Markov martingale with one-dimensional 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.