Abstract: We investigate the notion of geodesic for non anticipative processes.
More precisely, on a probability space endowed with a filtration, we look for an adapted real-valued process with fixed initial and final values, and which minimises the expected mean squared velocity. When the final time is some deterministic time, then the problem is easily solved; the solution can be written explicitly, and is characterised by the fact that the velocity should be a martingale. When the final time is a stopping time, we also can write an “explicit” solution, but it involves a change of probability.
Another problem is to investigate the notion of geodesic in the space of stopping times. In this case, the characterising property is that the squared velocity of the solution should be a martingale.
Several extensions of these two basic questions can be addressed: the case of manifold-valued processes, minimising the expected path length instead of the expected mean squared velocity, considering the multidimensional analogue of stopping times.