Mean-field BSDEs with jumps and dual representation for global risk measures

Rui Chen; Roxana Dumitrescu; Andreea Minca; Agnès Sulem

doi:10.3934/puqr.2023002

Article Contents

2023, Volume 8, Issue 1: 33-52. Doi: 10.3934/puqr.2023002

This issue Previous Article A universal robust limit theorem for nonlinear Lévy processes under sublinear expectation Next Article Ergodic switching control for diffusion-type processes

Mean-field BSDEs with jumps and dual representation for global risk measures

1.
INRIA Paris, 2 rue Simone Iff, CS 42112, 75589 Paris, Cedex 12, France
2.
Université Paris-Dauphine, 75775 Paris, Cedex 16, France
3.
Department of Mathematics, King’s College London, Strand, London WC2R 2LS, UK
4.
Department of Operations Research and Information Engineering, Cornell University, 222 Rhodes Hall, Ithaca, NY 14853, USA

Corresponding author: Agnès Sulem
Corresponding author: Agnès Sulem

Received: October 26, 2021

Accepted: October 04, 2022

Early access: November 2022

Published: March 2023

Abstract / Introduction Full Text(HTML) Related Papers Cited by

Abstract

We study mean-field BSDEs with jumps and a generalized mean-field operator that can capture higher-order interactions. We interpret the BSDE solution as a dynamic risk measure for a representative bank whose risk attitude is influenced by the system. This influence can come in a wide class of choices, including the average system state or average intensity of system interactions. Using Fenchel−Legendre transforms, our main result is a dual representation for the expectation of the risk measure in the convex case. In particular, we exhibit its dependence on the mean-field operator.

Keywords:

Mathematics Subject Classification: 60H10, 60H30.

Citation:

FullText(HTML)

Related Papers

Cited by

References

[1]	Agram, N., Hu, Y. and Øksendal, B., Mean-field backward stochastic differential equations and applications, Systems & Control Letters, 2022, 162: 105196.
[2]	Amini, H., Cao, Z. and Sulem, A., Graphon mean-field backward stochastic differential equations with jumps and associated dynamic risk measures, SSRN Electronic Journal, 2022, https://dx.doi.org/10.2139/ssrn.4162616.
[3]	Barrieu, P. and El Karoui, N., Optimal derivatives design under dynamic risk measures, In: Yin, G. and Zhang, Q.(eds.), Mathematics of Finance, Contemporary Mathematics, AMS, Providence, Rhode Island, 2004, 351: 13–26.
[4]	Bollobás, B., Janson, S. and Riordan, O., The phase transition in inhomogeneous random graphs, Random Structures & Algorithms, 2007, 31(1): 3−122.
[5]	Buckdahn, R., Djehiche, B., Li, J. and Peng, S., Mean-field backward stochastic differential equations: A limit approach, The Annals of Probability, 2009, 37(4): 1524−1565.
[6]	Carmona, R., Fouque, J.-P. and Sun, L.-H., Mean-field games and systemic risk, Communications in Mathematical Sciences, 2015, 13(4): 911−933. doi: 10.4310/CMS.2015.v13.n4.a4.
[7]	Chew, S. H., Epstein, L. G. and Segal, U., Mixture symmetry and quadratic utility, Econometrica, 1991, 59(1): 139−163. doi: 10.2307/2938244.
[8]	Cohn, D. l., Measure Theory, 2nd ed., Birkhäuser, New York, NY, 2013.
[9]	Craig, B. and Von Peter, G., Interbank tiering and money center banks, Journal of Financial Intermediation, 2014, 23(3): 322−347. doi: 10.1016/j.jfi.2014.02.003.
[10]	Dana, R.-A., A representation result for concave Schur concave functions, Mathematical Finance, 2005, 15(4): 613−634. doi: 10.1111/j.1467-9965.2005.00253.x.
[11]	Dellacherie, C. and Meyer, P.-A., Probabilité et Potentiel, Chap. I-IV, Hermann, 1975.
[12]	Ekeland, I. and Témam, R., Convex Analysis and Variational Problems, North Holland, Amsterdam, 1976.
[13]	Fouque, J.-P. and Ichiba, T., Stability in a model of interbank lending, SIAM Journal on Financial Mathematics, 2013, 4(1): 784−803. doi: 10.1137/110841096.
[14]	Frittelli, M. and Rosazza-Gianin, E., Dynamic convex risk measures, In: Szegö, G.(ed.), Risk Measures for the 21st Century, John Wiley & Sons, Hoboken, NJ, 2004: 227–248.
[15]	Garnier, J., Papanicolaou, G. and Yang, T.-W., Large deviations for a mean field model of systemic risk, SIAM Journal on Financial Mathematics, 2013, 4(1): 151−184. doi: 10.1137/12087387X.
[16]	Garnier, J., Papanicolaou, G. and Yang, T.-W., Diversification in financial networks may increase systemic risk, In: Fouque, J. and Langsam, J.(eds.), Handbook on Systemic Risk, Cambridge University Press, Cambridge, 2013: 432–443.
[17]	Hu, K., Ren, Z. and Yang, J. J., Principal-agent problem with multiple principals, arXiv: 1904.01413v2, 2019.
[18]	Laurière, M. and Tangpi, L., Backward propagation of chaos, Electron. J. Probab., 2022, 27: 1−30. doi: 10.48550/arXiv.1911.06835.
[19]	Laurière, M. and Tangpi, L., Convergence of large population games to mean field games with interaction through the controls, SIAM Journal on Mathematical Analysis, 2022, 54(3): 3535−3574. doi: 10.1137/22M1469328.
[20]	Li, J., Mean-field forward and backward SDEs with jumps and associated nonlocal quasi-linear integral-PDEs, Stochastic Processes and their Applications, 2018, 128(9): 3118−3180. doi: 10.1016/j.spa.2017.10.011.
[21]	Li, J. and Min, H., Controlled mean-field backward stochastic differential equations with jumps involving the value function, Journal of Systems Science and Complexity, 2016, 29(5): 1238−1268. doi: 10.1007/s11424-016-4275-5.
[22]	Peng, S., Nonlinear expectations, nonlinear evaluations and risk measures, In: Frittelli, M. and Runggaldier, W. (eds.), Stochastic Methods in Finance, Lecture Notes in Mathematics, Springer, Berlin, Heidelberg, 2004, 1856: 165–253.
[23]	Quenez, M.-C. and Sulem, A., BSDEs with jumps, optimization and applications to dynamic risk measures, Stochastic Processes and their Applications, 2013, 123(8): 3328−3357. doi: 10.1016/j.spa.2013.02.016.
[24]	Quenez, M.-C. and Sulem, A., Reflected BSDEs and robust optimal stopping for dynamic risk measures with jumps, Stochastic Processes and their Applications, 2014, 124(9): 3031−3054. doi: 10.1016/j.spa.2014.04.007.
[25]	Tang, S. J. and Li, X. J., Necessary conditions for optimal control of stochastic systems with random jumps, SIAM Journal on Control and Optimization, 1994, 32(5): 1447−1475. doi: 10.1137/S0363012992233858.