Does there exist a martingale which has Marginal distributions same as Brownian Motion marginals but the process itself not being Brownian motion? Any references are highly appreciated. Thanks.
This looks relevant.
Kais Hamza and Fima C. Klebaner, “A Family of Non-Gaussian Martingales with Gaussian Marginals,” Journal of Applied Mathematics and Stochastic Analysis, vol. 2007, Article ID 92723, 19 pages, 2007. doi:10.1155/2007/92723