Dynamic risk measuring: discrete time in a context of uncertainty, and continuous time on a probability space

We characterize time-consistent dynamic risk measures. In discrete time in context of uncertainty, we canonically associate a class of probability measures to any dynamic risk measure when the filtration comes from a process bounded at each time. Dynamic risk measures are conditional risk measures on a bigger space. In continuous time, we characterize time consistency, studying composition of conditional risk measures. Using sufficient conditions for time consistency, and BMO martingales, we construct new families of time-consistent dynamic risk measures. Some are continuous generalizing those coming from BSDE. Others are with jumps. keywords: dynamic risk measures, conditional risk measures, time consistency, uncertainty, BMO martingales JEL Classification: D81,D52. Mathematics Subject Classification (2000): 91B30, 91B70, 60G44, 28A20,46A20.