A construction is given for a martingale having a given right continuous strictly positive submartingale as its absolute value. The existence of a martingale whose absolute value has the same distribution as the given submartingale was shown by Gilat.
"Martingales with Given Absolute Value." Ann. Probab. 7 (6) 1056 - 1058, December, 1979. https://doi.org/10.1214/aop/1176994899