Time Consistent Dynamic Risk Processes, Cadlag Modification

Working in a continuous time setting, we extend to the general case of dynamic risk measures continuous from above the characterization of time consistency in terms of ``cocycle condition'' of the minimal penalty function. We prove also the supermartingale property for general time consistent dynamic risk measures. When the time consistent dynamic risk measure (continuous from above) is normalized and non degenerate, we prove, under a mild condition, that the dynamic risk process of any financial instrument has a cadlag modification. This condition is always satisfied in case of continuity from below.