Convex risk measures and the dynamics of their penalty functions

SUMMARY We study various properties of a dynamic convex risk measure for bounded random variables which describe the discounted terminal values of financial positions. In particular we characterize time-consistency by a joint supermartingale property of the risk measure and its penalty function. Moreover we discuss the limit behavior of the risk measure in terms of asymptotic safety and of asymptotic precision, a property which may be viewed as a non-linear analogue of martingale convergence. These results are illustrated by the entropic dynamic risk measure.

[1]  H. Föllmer,et al.  Optional decompositions under constraints , 1997 .

[2]  Philippe Artzner,et al.  Coherent Measures of Risk , 1999 .

[3]  F. Delbaen.  Coherent risk measures , 2000 .

[4]  F. Delbaen.  Coherent Risk Measures on General Probability Spaces , 2002 .

[5]  M. Frittelli,et al.  Putting order in risk measures , 2002 .

[6]  Frank Riedel,et al.  Dynamic Coherent Risk Measures , 2003 .

[7]  Marco Frittelli,et al.  Dynamic convex risk measures , 2004 .

[8]  Terry J. Lyons,et al.  Stochastic finance. an introduction in discrete time , 2004 .

[9]  F. Delbaen,et al.  Dynamic Monetary Risk Measures for Bounded Discrete-Time Processes , 2004, math/0410453.

[10]  Giacomo Scandolo,et al.  Conditional and dynamic convex risk measures , 2005, Finance Stochastics.

[11]  F. Delbaen.  The Structure of m–Stable Sets and in Particular of the Set of Risk Neutral Measures , 2006 .

[12]  Jocelyne Bion-Nadal.  Dynamic risk measuring: discrete time in a context of uncertainty, and continuous time on a probability space , 2006 .

[13]  S. Weber,et al.  DISTRIBUTION‐INVARIANT RISK MEASURES, INFORMATION, AND DYNAMIC CONSISTENCY , 2006 .

[14]  Jocelyne Bion-Nadal.  Bid-Ask Dynamic Pricing in Financial Markets with Transaction Costs and Liquidity Risk , 2007 .

[15]  Susanne Klöppel,et al.  DYNAMIC INDIFFERENCE VALUATION VIA CONVEX RISK MEASURES , 2007 .

[16]  Sina Tutsch.  Konsistente und konsequente dynamische Risikomaße und das Problem der Aktualisierung , 2007 .

[17]  David Heath,et al.  Coherent multiperiod risk adjusted values and Bellman’s principle , 2007, Ann. Oper. Res..

[18]  Jocelyne Bion-Nadal,et al.  Dynamic risk measures: Time consistency and risk measures from BMO martingales , 2008, Finance Stochastics.