Electronic Communications in Probability

Existence and Uniqueness of Solutions for BSDEs with Locally Lipschitz Coefficient

Khaled Bahlali

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We deal with multidimensional backward stochastic differential equations (BSDE) with locally Lipschitz coefficient in both variables $ y,z $ and an only square integrable terminal data. Let $ L_N $ be the Lipschitz constant of the coefficient on the ball $ B(0,N) $ of $ R^d\times R^{dr} $. We prove that if $ L_N = O (\sqrt {\log N }) $, then the corresponding BSDE has a unique solution. Moreover, the stability of the solution is established under the same assumptions. In the case where the terminal data is bounded, we establish the existence and uniqueness of the solution also when the coefficient has an arbitrary growth (in $ y $) and without restriction on the behaviour of the Lipschitz constant $ L_N $.

Article information

Electron. Commun. Probab., Volume 7 (2002), paper no. 17, 169-179.

Accepted: 5 August 2002
First available in Project Euclid: 16 May 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60H10: Stochastic ordinary differential equations [See also 34F05]

Backward stochastic differential equations (BSDE) locally Lipschitzfunction

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Bahlali, Khaled. Existence and Uniqueness of Solutions for BSDEs with Locally Lipschitz Coefficient. Electron. Commun. Probab. 7 (2002), paper no. 17, 169--179. doi:10.1214/ECP.v7-1058. https://projecteuclid.org/euclid.ecp/1463434784

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  • K. Bahlali (2001), Backward stochastic differential equations with locally Lipschitz coefficient. C.R.A.S Paris, serie I, 333, 481-486.
  • P. Briand, R. Carmona, (2000), BSDEs with polynomial growth generators. J. Appl. Math. Stochastic Anal. 13, 3, 207-238.
  • R. Darling, E. Pardoux, (1997), Backward SDE with monotonicity and random terminal time. Ann. of Probab. 25, 1135-1159.
  • A. Dermoune, S. Hamadène, Y. Ouknine, (1999), Backward stochastic differential equation with local time. Stoch. Stoch. Reports. 66, 103-119.
  • N. El Karoui, S. Peng, M.C. Quenez, (1997), Backward stochastic differential equations in finance. Math. Finance. 7, 1-71.
  • N. El-Karoui and S. Mazliak eds. Pitman Research Notes in Mathematics. Series 364.
  • S. Hamadène, (1996), Equations différentielles stochastiques rétrogrades: Le cas localement lipschitzien. Ann. Inst. Henri Poincaré. 32, 645-660.
  • S. Hamadène, (2000), Multidimensional Backward SDE's with Uniformly Continuous Coefficient. Monte Carlo 2000 conference at Monte Carlo, France, 3-5 jul. 2000, Preprint, Université du Maine.
  • S. Hamadène, J.P. Lepeletier, S. Peng, (1997), BSDE With continuous coefficients and applications to Markovian nonzero sumstochastic differential games. in N. El-Karoui and S. Mazliak eds. Pitman Research Notes in Mathematics. Series 364.
  • M. Kobylanski, (1997), Résultats d;existence et d'unicité pour des équations différentielles stochastiques rétrogrades avec des générateurs à croissance quadratique. C.R.A.S. Paris, Série 1 Math. 324, 81-86.
  • J.P. Lepeltier, J. San Martin, (1998), Existence for BSDE with Superlinear-Quadratic coefficients. Stoch. Stoch.Reports. 63, 227-240.
  • J. Ma, J. Yong (1999), Forward-Backward Stochastic Differential Equations and their applications. Lectures Note in Mathematics. 1702. Springer.
  • X. Mao, (1995), Adapted solutions of backward stochastic differential Equations with non-Lipschitz coefficient. Stoch. Proc. Appl. 58, 281-292.
  • E. Pardoux, (1999), BSDE's, weak convergence and homogenization of semilinear PDEs. In F. Clarke and R. Stern eds. Nonlin. Analy., Diff. Equa. and Control, Kluwer Acad. Publi. Dordrecht. 503-549.
  • E. Pardoux, S. Peng, (1990), Adapted solution of a backward stochastic differential equation. Syst. Cont. Letters. 14, 55-61.
  • E. Pardoux, S. Peng, (1992), Backward SDEs and quasilinear PDEs. In Stochastic Partial Differential Equations and their Applications. B.L. Rozovskii and R. Sowers, eds. Lecture Notes and inform. Sci. 176, 200-217.
  • S. Peng, (1991), Probabilistic interpretation for systems of quasilinear parabolic partial differential equations. Stoch. Stoch. Reports. 37, 61-74.
  • R. Situ (1997), On solutions of backward stochastic differential equations with jumps and applications. Stoch. Process. Appl. 66, 209–236.