Let $\varphi$ be a convex function. A sufficient condition is given that a submartingale is equal in law to the $\varphi$-image of a martingale. It follows that each nonnegative submartingale, without any assumption on the regularity of the paths, can be obtained as the absolute value of a martingale.
"Martingales with Given Convex Image." Ann. Probab. 10 (4) 1085 - 1087, November, 1982. https://doi.org/10.1214/aop/1176993733