Finance and Stochastics

, Volume 12, Issue 2, pp 219–244 | Cite as

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

  • Jocelyne Bion-Nadal


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 BMO martingales, we provide new classes of time-consistent dynamic risk measures. These families generalize those obtained from backward stochastic differential equations. Quite importantly, starting with right-continuous BMO martingales, this construction naturally leads to paths with jumps.


Dynamic risk measures Conditional risk measures Time consistency BMO martingales 

Mathematics Subject Classification (2000)

91B30 91B70 60G44 28A20 46A20 


D81 D52 C61 


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Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Centre de Mathématiques Appliquées (CMAP, UMR CNRS 7641)Ecole PolytechniquePalaiseau cedexFrance

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