A continuous non-Brownian motion martingale with Brownian motion marginal distributions

Journal article, 2008

This note exhibits a continuous martingale $M$ which is not Brownian motion, but has the same univariate marginal distributions as Brownian motion. It is given by $M(t)=X_1(t)X_2(t)Y$, where $X_1$ and $X_2$ are independent copies of the diffusion $dX(t)=dB(t)(2X(t))^{-1},\ X(0)=0$, and $Y$ is an independent random variable with known density on $(0,\sqrt{2})$. The existence of such a martingale was an open problem until now.