The Annals of Probability

A class of globally solvable Markovian quadratic BSDE systems and applications

Hao Xing and Gordan Žitković

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


We establish existence and uniqueness for a wide class of Markovian systems of backward stochastic differential equations (BSDE) with quadratic nonlinearities. This class is characterized by an abstract structural assumption on the generator, an a priori local-boundedness property, and a locally-Hölder-continuous terminal condition. We present easily verifiable sufficient conditions for these assumptions and treat several applications, including stochastic equilibria in incomplete financial markets, stochastic differential games and martingales on Riemannian manifolds.

Article information

Ann. Probab., Volume 46, Number 1 (2018), 491-550.

Received: March 2016
Revised: March 2017
First available in Project Euclid: 5 February 2018

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G44: Martingales with continuous parameter 60G99: None of the above, but in this section 60H30: Applications of stochastic analysis (to PDE, etc.)
Secondary: 58J65: Diffusion processes and stochastic analysis on manifolds [See also 35R60, 60H10, 60J60] 91A15: Stochastic games 91B51: Dynamic stochastic general equilibrium theory

BSDE backward stochastic differential equations systems of BSDE quadratic nonlinearities stochastic equilibrium martingales on manifolds nonzero-sum stochastic games


Xing, Hao; Žitković, Gordan. A class of globally solvable Markovian quadratic BSDE systems and applications. Ann. Probab. 46 (2018), no. 1, 491--550. doi:10.1214/17-AOP1190.

Export citation


  • [1] Aronson, D. G. (1967). Bounds for the fundamental solution of a parabolic equation. Bull. Amer. Math. Soc. 73 890–896.
  • [2] Aubin, T. (1998). Some Nonlinear Problems in Riemannian Geometry. Springer Monographs in Mathematics. Springer, Berlin.
  • [3] Bally, V. and Matoussi, A. (2001). Weak solutions for SPDEs and backward doubly stochastic differential equations. J. Theoret. Probab. 14 125–164.
  • [4] Barles, G. and Lesigne, E. (1997). SDE, BSDE and PDE. In Backward Stochastic Differential Equations (Paris, 19951996). Pitman Res. Notes Math. Ser. 364 47–80. Longman, Harlow.
  • [5] Barrieu, P. and El Karoui, N. (2013). Monotone stability of quadratic semimartingales with applications to unbounded general quadratic BSDEs. Ann. Probab. 41 1831–1863.
  • [6] Bensoussan, A. and Frehse, J. (2000). Stochastic games for $N$ players. J. Optim. Theory Appl. 105 543–565. Special Issue in honor of Professor David G. Luenberger.
  • [7] Bensoussan, A. and Frehse, J. (2002). Smooth solutions of systems of quasilinear parabolic equations. ESAIM Control Optim. Calc. Var. 8 169–193.
  • [8] Bismut, J.-M. (1973). Conjugate convex functions in optimal stochastic control. J. Math. Anal. Appl. 44 384–404.
  • [9] Blache, F. (2005). Backward stochastic differential equations on manifolds. Probab. Theory Related Fields 132 391–437.
  • [10] Blache, F. (2006). Backward stochastic differential equations on manifolds. II. Probab. Theory Related Fields 136 234–262.
  • [11] Briand, P. and Elie, R. (2013). A simple constructive approach to quadratic BSDEs with or without delay. Stochastic Process. Appl. 123 2921–2939.
  • [12] Briand, P. and Hu, Y. (2006). BSDE with quadratic growth and unbounded terminal value. Probab. Theory Related Fields 136 604–618.
  • [13] Briand, P. and Hu, Y. (2008). Quadratic BSDEs with convex generators and unbounded terminal conditions. Probab. Theory Related Fields 141 543–567.
  • [14] Çetin, U. and Danilova, A. (2016). Markovian Nash equilibrium in financial markets with asymmetric information and related forward-backward systems. Ann. Appl. Probab. 26 1996–2029.
  • [15] Chang, K.-C., Ding, W. Y. and Ye, R. (1992). Finite-time blow-up of the heat flow of harmonic maps from surfaces. J. Differential Geom. 36 507–515.
  • [16] Cheridito, P., Horst, U., Kupper, M. and Pirvu, T. A. (2016). Equilibrium pricing in incomplete markets under translation invariant preferences. Math. Oper. Res. 41 174–195.
  • [17] Cheridito, P. and Nam, K. (2014). BSDEs with terminal conditions that have bounded Malliavin derivative. J. Funct. Anal. 266 1257–1285.
  • [18] Cheridito, P. and Nam, K. (2015). Multidimensional quadratic and subquadratic BSDEs with special structure. Stochastics 87 871–884.
  • [19] Chitashvili, R. and Mania, M. (1997). On functions transforming a Wiener process into a semimartingale. Probab. Theory Related Fields 109 57–76.
  • [20] Choi, J. H. and Larsen, K. (2015). Taylor approximation of incomplete Radner equilibrium models. Finance Stoch. 19 653–679.
  • [21] Darling, R. W. R. (1995). Constructing gamma-martingales with prescribed limit, using backwards SDE. Ann. Probab. 23 1234–1261.
  • [22] Davis, C. (1954). Theory of positive linear dependence. Amer. J. Math. 76 733–746.
  • [23] Delarue, F. (2003). Estimates of the solutions of a system of quasi-linear PDEs. A probabilistic scheme. In Séminaire de Probabilités XXXVII. Lecture Notes in Math. 1832 290–332. Springer, Berlin.
  • [24] Delbaen, F., Hu, Y. and Bao, X. (2011). Backward SDEs with superquadratic growth. Probab. Theory Related Fields 150 145–192.
  • [25] Eells, J. Jr. and Sampson, J. H. (1964). Harmonic mappings of Riemannian manifolds. Amer. J. Math. 86 109–160.
  • [26] El-Karoui, N. and Hamadène, S. (2003). BSDEs and risk-sensitive control, zero-sum and nonzero-sum game problems of stochastic functional differential equations. Stochastic Process. Appl. 107 145–169.
  • [27] El Karoui, N., Peng, S. and Quenez, M. C. (1997). Backward stochastic differential equations in finance. Math. Finance 7 1–71.
  • [28] Émery, M. (1989). Stochastic Calculus in Manifolds. Springer, Berlin.
  • [29] Espinosa, G.-E. and Touzi, N. (2015). Optimal investment under relative performance concerns. Math. Finance 25 221–257.
  • [30] Evans, L. C. (1998). Partial Differential Equations. Graduate Studies in Mathematics 19. Amer. Math. Soc., Providence, RI.
  • [31] Frehse, J. (1988). Remarks on diagonal elliptic systems. In Partial Differential Equations and Calculus of Variations. Lecture Notes in Math. 1357 198–210. Springer, Berlin.
  • [32] Frei, C. (2014). Splitting multidimensional BSDEs and finding local equilibria. Stochastic Process. Appl. 124 2654–2671.
  • [33] Frei, C. and dos Reis, G. (2011). A financial market with interacting investors: Does an equilibrium exist? Math. Financ. Econ. 4 161–182.
  • [34] Friedman, A. (1964). Partial Differential Equations of Parabolic Type. Prentice-Hall, Englewood Cliffs, NJ.
  • [35] Hsu, E. P. (2002). Stochastic Analysis on Manifolds. Graduate Studies in Mathematics 38. Amer. Math. Soc., Providence, RI.
  • [36] Hu, Y. and Peng, S. (2006). On the comparison theorem for multidimensional BSDEs. C. R. Math. Acad. Sci. Paris 343 135–140.
  • [37] Hu, Y. and Tang, S. (2016). Multi-dimensional backward stochastic differential equations of diagonally quadratic generators. Stochastic Process. Appl. 126 1066–1086.
  • [38] Il’in, A. M., Kalašnikov, A. S. and Oleĭnik, O. A. (1962). Second-order linear equations of parabolic type. Uspekhi Mat. Nauk 17 3–146.
  • [39] John, F. (1978). Partial Differential Equations, 3rd ed. Applied Mathematical Sciences 1. Springer, New York.
  • [40] Kardaras, C., Xing, H. and Žitković (2015). Incomplete stochastic equilibria with exponential utilities: Close to Pareto optimality. Working paper.
  • [41] Kendall, W. S. (1990). Probability, convexity, and harmonic maps with small image. I. Uniqueness and fine existence. Proc. Lond. Math. Soc. (3) 61 371–406.
  • [42] Kobylanski, M. (2000). Backward stochastic differential equations and partial differential equations with quadratic growth. Ann. Probab. 28 558–602.
  • [43] Kramkov, D. and Pulido, S. (2016). A system of quadratic BSDEs arising in a price impact model. Ann. Appl. Probab. 26 794–817.
  • [44] Ladyženskaja, O. A., Solonnikov, V. A. and Ural’ceva, N. N. (1968). Linear and Quasilinear Equations of Parabolic Type. Translations of Mathematical Monographs 23. Amer. Math. Soc., Providence, RI. Translated from the Russian by S. Smith.
  • [45] Lejay, A. (2002). BSDE driven by Dirichlet process and semi-linear parabolic PDE. Application to homogenization. Stochastic Process. Appl. 97 1–39.
  • [46] Lepeltier, J. P. and San Martin, J. (1997). Backward stochastic differential equations with continuous coefficient. Statist. Probab. Lett. 32 425–430.
  • [47] Lieberman, G. M. (1996). Second Order Parabolic Differential Equations. World Scientific, River Edge, NJ.
  • [48] Matoussi, A. and Xu, M. (2008). Sobolev solution for semilinear PDE with obstacle under monotonicity condition. Electron. J. Probab. 13 1035–1067.
  • [49] Pardoux, É. and Peng, S. (1992). Backward stochastic differential equations and quasilinear parabolic partial differential equations. In Stochastic Partial Differential Equations and Their Applications (Charlotte, NC, 1991). Lect. Notes Control Inf. Sci. 176 200–217. Springer, Berlin.
  • [50] Pardoux, É. and Peng, S. G. (1990). Adapted solution of a backward stochastic differential equation. Systems Control Lett. 14 55–61.
  • [51] Peng, S. (1999). Open problems on backward stochastic differential equations. In Control of Distributed Parameter and Stochastic Systems (Hangzhou, 1998) 265–273. Kluwer Academic, Boston, MA.
  • [52] Stroock, D. W. and Varadhan, S. R. S. (2006). Multidimensional Diffusion Processes. Springer, Berlin. Reprint of the 1997 edition.
  • [53] Struwe, M. (1981). On the Hölder continuity of bounded weak solutions of quasilinear parabolic systems. Manuscripta Math. 35 125–145.
  • [54] Subrahmanyam, A. (1991). Risk aversion, market liquidity, and price efficiency. Rev. Financ. Stud. 4 417–441.
  • [55] Tang, S. (2003). General linear quadratic optimal stochastic control problems with random coefficients: Linear stochastic Hamilton systems and backward stochastic Riccati equations. SIAM J. Control Optim. 42 53–75.
  • [56] Tevzadze, R. (2008). Solvability of backward stochastic differential equations with quadratic growth. Stochastic Process. Appl. 118 503–515.
  • [57] Udrişte, C. (1994). Convex Functions and Optimization Methods on Riemannian Manifolds. Mathematics and Its Applications 297. Kluwer Academic, Dordrecht.
  • [58] Widman, K.-O. (1971). Hölder continuity of solutions of elliptic systems. Manuscripta Math. 5 299–308.
  • [59] Zhao, Y. (2012). Stochastic equilibria in a general class of incomplete Brownian market environments. Ph.D. thesis, Univ. Texas at Austin.
  • [60] Žitković, G. (2012). An example of a stochastic equilibrium with incomplete markets. Finance Stoch. 16 177–206.