Dynamic risk measures: Time consistency and risk measures from BMO martingales

Time consistency is a crucial property for dynamic risk measures. Making use of the dual representation for conditional risk measures, we characterize the time consistency by a cocycle condition for the minimal penalty function. Taking advantage of this cocycle condition, we introduce a new methodology for the construction of time-consistent dynamic risk measures. Starting with BMOmartingales, we provide new classes of time-consistent dynamic risk measures. These families generalize the Backward Stochastic Differential Equations. Quite importantly, starting with right continuous BMO martingales this construction leads naturally to paths with jumps. keywords dynamic risk measures, conditional risk measures, time consistency, BMO martingales