We show that every discrete-time martingale can be interpolated to give a continuous, continuous-time martingale, and provide a necessary and sufficient condition for the existence of an interpolated martingale with no flat spots.
"Interpolation of Martingales." Ann. Probab. 5 (5) 804 - 806, October, 1977. https://doi.org/10.1214/aop/1176995723