A class of time inconsistent risk measures and backward stochastic Volterra integral equations

In this paper we derive a new class of time inconsistent dynamic coherent risk measures, allowing the interest rates involved to be random, which extends and modifies the results on risk measures in Appl. Anal. 86 (2007), 1429–1442. Since in this procedure one important mathematical tool is a kind of backward stochastic Volterra integral equations (BSVIEs in short) with stochastic Lipschitz coefficients, this paper firstly is devoted to develop the existence and uniqueness theory for the corresponding BSVIEs, which not only generalizes the results in Stoch. Proc. Appl. 116 (2006), 779–795; Appl. Anal. 86 (2007), 1429–1442; Probab. Theory Related Fields 142 (2008), 21–77; but also simplifies considerably the related arguments there.

[1]  Yufeng Shi,et al.  Symmetrical Solutions of Backward Stochastic Volterra Integral Equations and Their Applications , 2009, 0910.5580.

[2]  Jiongmin Yong,et al.  Regularity of Backward Stochastic Volterra Integral Equations in Hilbert Spaces , 2010 .

[3]  I. Ekeland,et al.  Investment and consumption without commitment , 2007, 0708.0588.

[4]  J. Bion-Nadal.  Time consistent dynamic risk processes , 2009 .

[5]  Xicheng Zhang,et al.  NON-LIPSCHITZ BACKWARD STOCHASTIC VOLTERRA TYPE EQUATIONS WITH JUMPS , 2007 .

[6]  Jianzhong Lin,et al.  Adapted solution of a backward stochastic nonlinear Volterra integral equation , 2002 .

[7]  Jiongmin Yong,et al.  Continuous-time dynamic risk measures by backward stochastic Volterra integral equations , 2007 .

[8]  Yong Ren,et al.  On Solutions of Backward Stochastic Volterra Integral Equations with Jumps in Hilbert Spaces , 2010 .

[9]  Emanuela Rosazza Gianin,et al.  Risk measures via g-expectations , 2006 .

[10]  R. H. Strotz.  Myopia and Inconsistency in Dynamic Utility Maximization , 1955 .

[11]  Alexander Schied,et al.  Convex measures of risk and trading constraints , 2002, Finance Stochastics.

[12]  David I. Laibson,et al.  Golden Eggs and Hyperbolic Discounting , 1997 .

[13]  Jiongmin Yong,et al.  Well-posedness and regularity of backward stochastic Volterra integral equations , 2008 .

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

[15]  Jiongmin Yong,et al.  Backward stochastic Volterra integral equations and some related problems , 2006 .

[16]  Yufeng Shi,et al.  Mean-Field Backward Stochastic Volterra Integral Equations , 2011, ArXiv.

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

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