Electronic Journal of Probability

A stability approach for solving multidimensional quadratic BSDEs

Jonathan Harter and Adrien Richou

Full-text: Open access

Abstract

We establish an existence and uniqueness result for a class of multidimensional quadratic backward stochastic differential equations (BSDE). This class is characterized by constraints on some uniform a priori estimate on solutions of a sequence of approximated BSDEs. We also present effective examples of applications. Our approach relies on the strategy developed by Briand and Elie in [Stochastic Process. Appl. 123 2921–2939] concerning scalar quadratic BSDEs.

Article information

Source
Electron. J. Probab., Volume 24 (2019), paper no. 4, 51 pp.

Dates
Received: 9 March 2018
Accepted: 24 December 2018
First available in Project Euclid: 8 February 2019

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1549616425

Digital Object Identifier
doi:10.1214/18-EJP260

Subjects
Primary: 65C30: Stochastic differential and integral equations

Keywords
BSDES quadratic BSDES martingales

Rights
Creative Commons Attribution 4.0 International License.

Citation

Harter, Jonathan; Richou, Adrien. A stability approach for solving multidimensional quadratic BSDEs. Electron. J. Probab. 24 (2019), paper no. 4, 51 pp. doi:10.1214/18-EJP260. https://projecteuclid.org/euclid.ejp/1549616425


Export citation

References

  • [Arn97] Marc Arnaudon. Differentiable and analytic families of continuous martingales in manifolds with connection. Probability Theory and Related Fields, 108(2):219–257, June 1997.
  • [BB88] Rodrigo Bañuelos and Andrew Bennett. Paraproducts and commutators of martingale transforms. Proceedings of the American Mathematical Society, 103(4):1226–1234, 1988.
  • [BE13] Philippe Briand and Romuald Elie. A simple constructive approach to quadratic BSDEs with or without delay. Stochastic Processes and their Applications, 123(8):2921–2939, August 2013.
  • [BEK13] Pauline Barrieu and Nicole El Karoui. Monotone stability of quadratic semimartingales with applications to unbounded general quadratic BSDEs. The Annals of Probability, 41(3B):1831–1863, May 2013.
  • [BH06] Philippe Briand and Ying Hu. BSDE with quadratic growth and unbounded terminal value. Probability Theory and Related Fields, 136(4):604–618, April 2006.
  • [Bis73] Jean-Michel Bismut. Conjugate convex functions in optimal stochastic control. Journal of Mathematical Analysis and Applications, 44(2):384–404, November 1973.
  • [BLDR15] Jana Bielagk, Arnaud Lionnet, and Gonçalo Dos Reis. Equilibrium pricing under relative performance concerns. arXiv.org, November 2015.
  • [CM13] Besik Chikvinidze and Misha Mania. New Proofs of Some Results on Bounded Mean Oscillation Martingales Using Backward Stochastic Differential Equations. Journal of Theoretical Probability, 27(4):1213–1228, October 2013.
  • [CN15] Patrick Cheridito and Kihun Nam. Multidimensional quadratic and subquadratic BSDEs with special structure. 87(5):871–884, April 2015.
  • [Dar95] Richard W. R Darling. Constructing gamma-martingales with prescribed limit, using backwards SDE. The Annals of Probability, 23(3):1234–1261, 1995.
  • [DHR11] Freddy Delbaen, Ying Hu, and Adrien Richou. On the uniqueness of solutions to quadratic BSDEs with convex generators and unbounded terminal conditions. Annales de l’Institut Henri Poincaré. Probabilités et Statistiques, 47(2):559–574, May 2011.
  • [DT08] Freddy Delbaen and Shanjian Tang. Harmonic analysis of stochastic equations and backward stochastic differential equations. Probability Theory and Related Fields, 146(1-2):291–336, December 2008.
  • [EKH03] Nicole El Karoui and Said Hamadène. BSDEs and risk-sensitive control, zero-sum and nonzero-sum game problems of stochastic functional differential equations. Stochastic Processes and their Applications, 107(1):145–169, September 2003.
  • [EKPQ97] Nicole El Karoui, Shige Peng, and Marie-Claire Quenez. Backward Stochastic Differential Equations in Finance. Mathematical Finance, 7(1):1–71, January 1997.
  • [Eme89] Michel Emery. Stochastic calculus in manifolds. Universitext. Springer-Verlag, Berlin, Berlin, Heidelberg, 1989.
  • [EP16] Romuald Elie and Dylan Possamaï. Contracting theory with competitive interacting agents. arXiv.org, page arXiv:1605.08099, May 2016.
  • [ET15] Gilles Edouard Espinosa and Nizar Touzi. Optimal investment under relative performance concerns. Mathematical Finance, 25(2):221–257, April 2015.
  • [FDR11] Christoph Frei and Gonçalo Dos Reis. A financial market with interacting investors: does an equilibrium exist? Mathematics and Financial Economics, 4(3):161–182, February 2011.
  • [Fre14] Christoph Frei. Splitting multidimensional BSDEs and finding local equilibria. Stochastic Processes and their Applications, 124(8):2654–2671, August 2014.
  • [GY14] Stefan Geiss and Juha Ylinen. Decoupling on the Wiener Space, Related Besov Spaces, and Applications to BSDEs. arXiv.org, page arXiv:1409.5322, September 2014.
  • [HT16] Ying Hu and Shanjian Tang. Multi-dimensional backward stochastic differential equations of diagonally quadratic generators. Stochastic Processes and their Applications, 126(4):1066–1086, April 2016.
  • [JKL14] Asgar Jamneshan, Michael Kupper, and Peng Luo. Multidimensional quadratic BSDEs with separated generators. arXiv.org, December 2014.
  • [Kaz94] Norihiko Kazamaki. Continuous exponential martingales and BMO, volume 1579 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1994.
  • [Ken90] Wilfrid S Kendall. Probability, Convexity, and Harmonic Maps with Small Image I: Uniqueness and Fine Existence. Proceedings of the London Mathematical Society, s3-61(2):371–406, September 1990.
  • [Ken92] Wilfrid S Kendall. The Propeller: A Counterexample to a Conjectured Criterion for the Existence of Certain Convex Functions. Journal of the London Mathematical Society. Second Series, s2-46(2):364–374, October 1992.
  • [Kob00] Magdalena Kobylanski. Backward stochastic differential equations and partial differential equations with quadratic growth. Annals of Probability, 2000.
  • [KP16a] Dmitry Kramkov and Sergio Pulido. A system of quadratic BSDEs arising in a price impact model. The Annals of Applied Probability, 26(2):794–817, April 2016.
  • [KP16b] Dmitry Kramkov and Sergio Pulido. Stability and Analytic Expansions of Local Solutions of Systems of Quadratic BSDEs with Applications to a Price Impact Model. SIAM J. Financial Math., 7(1):567–587, 2016.
  • [KXŽ15] Constantinos Kardaras, Hao Xing, and Gordan Žitković. Incomplete stochastic equilibria with exponential utilities close to Pareto optimality. arXiv.org, page arXiv:1505.07224, May 2015.
  • [LT15] Peng Luo and Ludovic Tangpi. Solvability of coupled FBSDEs with quadratic and superquadratic growth. arXiv.org, page arXiv:1505.01796, May 2015.
  • [MR16] Carlo Marinelli and Michael Röckner. On the maximal inequalities of Burkholder, Davis and Gundy. Expositiones Mathematicae, 34(1):1–26, 2016.
  • [Nua06] David Nualart. The Malliavin calculus and related topics. Probability and its Applications (New York). Springer-Verlag, Berlin, second edition, 2006.
  • [Pic94] Jean Picard. Barycentres et martingales sur une variété. Annales de l’Institut Henri Poincaré. Probabilités et Statistiques, 30(4):647–702, 1994.
  • [PP90] Etienne Pardoux and Shige Peng. Adapted solution of a backward stochastic differential equation. Systems and Control Letters, 14(1):55–61, January 1990.
  • [Sch96] Walter Schachermayer. A characterisation of the closure of ${\mathcal H}^\infty $ in $BMO$. Séminaire de Probabilités de Strasbourg, 1996.
  • [Tev08] Revaz Tevzadze. Solvability of backward stochastic differential equations with quadratic growth. Stochastic Processes and their Applications, 118(3):503–515, March 2008.
  • [XŽ16] Hao Xing and Gordan Žitković. A class of globally solvable Markovian quadratic BSDE systems and applications. arXiv.org, page arXiv:1603.00217, March 2016.