The Annals of Probability

BSE’s, BSDE’s and fixed-point problems

Patrick Cheridito and Kihun Nam

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


In this paper, we introduce a class of backward stochastic equations (BSEs) that extend classical BSDEs and include many interesting examples of generalized BSDEs as well as semimartingale backward equations. We show that a BSE can be translated into a fixed-point problem in a space of random vectors. This makes it possible to employ general fixed-point arguments to establish the existence of a solution. For instance, Banach’s contraction mapping theorem can be used to derive general existence and uniqueness results for equations with Lipschitz coefficients, whereas Schauder-type fixed-point arguments can be applied to non-Lipschitz equations. The approach works equally well for multidimensional as for one-dimensional equations and leads to results in several interesting cases such as equations with path-dependent coefficients, anticipating equations, McKean–Vlasov-type equations and equations with coefficients of superlinear growth.

Article information

Ann. Probab., Volume 45, Number 6A (2017), 3795-3828.

Received: August 2015
Revised: September 2016
First available in Project Euclid: 27 November 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60H10: Stochastic ordinary differential equations [See also 34F05] 47H10: Fixed-point theorems [See also 37C25, 54H25, 55M20, 58C30]

Backward stochastic equation backward stochastic differential equation path-dependent coefficients anticipating equations McKean–Vlasov-type equations coefficients of superlinear growth


Cheridito, Patrick; Nam, Kihun. BSE’s, BSDE’s and fixed-point problems. Ann. Probab. 45 (2017), no. 6A, 3795--3828. doi:10.1214/16-AOP1149.

Export citation


  • Aliprantis, C. D. and Border, K. C. (2006). Infinite Dimensional Analysis: A Hitchhiker’s Guide, 3rd ed. Springer, Berlin.
  • Briand, P. and Hu, Y. (2006). BSDE with quadratic growth and unbounded terminal value. Probab. Theory Related Fields 136 604–618.
  • Briand, P. and Hu, Y. (2008). Quadratic BSDEs with convex generators and unbounded terminal conditions. Probab. Theory Related Fields 141 543–567.
  • Buckdahn, R., Li, J. and Peng, S. (2009). Mean-field backward stochastic differential equations and related partial differential equations. Stochastic Process. Appl. 119 3133–3154.
  • Cheridito, P. and Nam, K. (2015). Multidimensional quadratic and subquadratic BSDEs with special structure. Stochastics 87 871–884.
  • Chitasvili, R. J. (1983). Martingale ideology in the theory of controlled stochastic processes. In Probability Theory and Mathematical Statistics (Tbilisi, 1982). Lecture Notes in Math. 1021 73–92. Springer, Berlin.
  • Da Prato, G. (2006). An Introduction to Infinite-Dimensional Analysis. Springer, Berlin.
  • Delbaen, F., Hu, Y. and Richou, A. (2011). On the uniqueness of solutions to quadratic BSDEs with convex generators and unbounded terminal conditions. Ann. Inst. Henri Poincaré Probab. Stat. 47 559–574.
  • Delong, L. and Imkeller, P. (2010). Backward stochastic differential equations with time delayed generators—results and counterexamples. Ann. Appl. Probab. 20 1512–1536.
  • Delong, Ł. and Imkeller, P. (2010). On Malliavin’s differentiability of BSDEs with time delayed generators driven by Brownian motions and Poisson random measures. Stochastic Process. Appl. 120 1748–1775.
  • Frei, C. and dos Reis, G. (2011). A financial market with interacting investors: Does an equilibrium exist? Math. Financ. Econ. 4 161–182.
  • Ikeda, N. and Watanabe, S. (1989). Stochastic Differential Equations and Diffusion Processes, 2nd ed. North-Holland, Amsterdam.
  • Jacod, J. (1979). Calcul Stochastique et Problèmes de Martingales. Lectures Notes in Mathematics 714. Springer, Heidelberg.
  • Kobylanski, M. (2000). Backward stochastic differential equations and partial differential equations with quadratic growth. Ann. Probab. 28 558–602.
  • Krasnoselskii, M. A. (1964). Topological Methods in the Theory of Nonlinear Integral Equations. Macmillan, New York.
  • Liang, G., Lyons, T. and Qian, Z. (2011). Backward stochastic dynamics on a filtered probability space. Ann. Probab. 39 1422–1448.
  • Mania, M. and Tevzadze, R. (2003). A semimartingale backward equation and the variance-optimal martingale measure under general information flow. SIAM J. Control Optim. 42 1703–1726.
  • Pardoux, É. and Peng, S. G. (1990). Adapted solution of a backward stochastic differential equation. Systems Control Lett. 14 55–61.
  • Peng, S. (1999). Open problems on backward stochastic differential equations. In Control of Distributed Parameter and Stochastic Systems. IFIP Advances in Information and Communication Technology 13 265–273. Kluwer Academic, Boston, MA.
  • Peng, S. and Yang, Z. (2009). Anticipated backward stochastic differential equations. Ann. Probab. 37 877–902.
  • Smart, D. R. (1974). Fixed Point Theorems. Cambridge Tracts in Mathematics 66. Cambridge Univ. Press, Cambridge.
  • Tang, S. and Li, X. (1994). Necessary conditions for optimal control of stochastic systems with random jumps. SIAM J. Control Optim. 32 1447–1475.
  • Tevzadze, R. (2008). Solvability of backward stochastic differential equations with quadratic growth. Stochastic Process. Appl. 118 503–515.