In this paper we give the local time for some continuous two-parameter martingales with respect to the quadratic variation $\langle M\rangle$ and we study some of their sample path properties.
Marta Sanz. "Local Time for Two-Parameter Continuous Martingales with Respect to the Quadratic Variation." Ann. Probab. 16 (2) 778 - 792, April, 1988. https://doi.org/10.1214/aop/1176991787