Mean Field BSDEs and Global Dynamic Risk Measures

We study Mean-field BSDEs with jumps and a generalized mean-field operator that can capture higher order interactions such as those occurring on an inhomogeneous random graph. We provide comparison and strict comparison results. Based on these, we interpret the BSDE solution as a global dynamic risk measure that can account for the intensity of system interactions and therefore incorporate systemic risk. Using Fenchel-Legendre transforms, we establish a dual representation for the risk measure, and in particular we exhibit its dependence on the mean-field operator.

[1]  Juan Li,et al.  Controlled mean-field backward stochastic differential equations with jumps involving the value function , 2016, J. Syst. Sci. Complex..

[2]  Marco Frittelli,et al.  A unified approach to systemic risk measures via acceptance sets , 2015, 1503.06354.

[3]  Stefan Weber,et al.  Measures of Systemic Risk , 2015, SIAM J. Financial Math..

[4]  Jean-Pierre Fouque,et al.  Stability in a Model of Interbank Lending , 2013, SIAM J. Financial Math..

[5]  M. Quenez,et al.  BSDEs with jumps, optimization and applications to dynamic risk measures , 2013 .

[6]  R. Carmona,et al.  Mean Field Games and Systemic Risk , 2013, 1308.2172.

[7]  M. Quenez,et al.  Reflected BSDEs and robust optimal stopping for dynamic risk measures with jumps , 2012, 1212.6744.

[8]  Josselin Garnier,et al.  Large Deviations for a Mean Field Model of Systemic Risk , 2012, SIAM J. Financial Math..

[9]  Goetz von Peter,et al.  Interbank Tiering and Money Center Banks , 2010, SSRN Electronic Journal.

[10]  Juan Li,et al.  Mean-field backward stochastic differential equations: A limit approach , 2007, 0711.2162.

[11]  S. Peng,et al.  Mean-field backward stochastic differential equations and related partial differential equations , 2007, 0711.2167.

[12]  M. Royer.  Backward stochastic differential equations with jumps and related non-linear expectations , 2006 .

[13]  B. Bollobás,et al.  The phase transition in inhomogeneous random graphs , 2005, Random Struct. Algorithms.

[14]  Xunjing Li,et al.  Necessary Conditions for Optimal Control of Stochastic Systems with Random Jumps , 1994 .

[15]  Daniel H. Wagner.  Survey of Measurable Selection Theorems , 1977 .

[16]  S. Hamadène,et al.  REFLECTED BACKWARD STOCHASTIC DIFFERENTIAL EQUATION WITH JUMPS AND RANDOM OBSTACLE , 2018 .

[17]  G. Papanicolaou,et al.  Handbook on Systemic Risk: Diversification in Financial Networks may Increase Systemic Risk , 2013 .

[18]  N. El Karoui,et al.  Numerical Methods in Finance: Reflected Backward SDEs and American Options , 1997 .