We study the solution of one-dimensional generalized backward stochastic differential equation driven by Teugels martingales and an independent Brownian motion. We prove existence and uniqueness of the solution when the coefficient verifies some conditions of Lipschitz. If the coefficient is left continuous, increasing, and bounded, we prove the existence of a solution.
K. Bahlali, M. Eddahbi, and E. Essaky, “BSDE associated with Lévy processes and application to PDIE,” Journal of Applied Mathematics and Stochastic Analysis, vol. 16, no. 1, pp. 1–17, 2003.
View at: Google Scholar | Zentralblatt MATH | MathSciNetK. Bahlali and E. Pardoux, “Backward stochastic differential equations with locally monotone coefficient,” preprint, 2001.
View at: Google ScholarG. Barles, R. Buckdahn, and E. Pardoux, “Backward stochastic differential equations and integral-partial differential equations,” Stochastics and Stochastics Reports, vol. 60, no. 1-2, pp. 57–83, 1997.
View at: Google Scholar | Zentralblatt MATH | MathSciNetJ.-M. Bismut, “Conjugate convex functions in optimal stochastic control,” Journal of Mathematical Analysis and Applications, vol. 44, pp. 384–404, 1973.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNetB. Boufoussi and Y. Ouknine, “On a SDE driven by a fractional Brownian motion and with monotone drift,” Electronic Communications in Probability, vol. 8, pp. 122–134, 2003.
View at: Google Scholar | Zentralblatt MATH | MathSciNetPh. Briand and R. Carmona, “BSDEs with polynomial growth generators,” Journal of Applied Mathematics and Stochastic Analysis, vol. 13, no. 3, pp. 207–238, 2000.
View at: Google Scholar | Zentralblatt MATH | MathSciNetR. W. R. Darling and E. Pardoux, “Backwards SDE with random terminal time and applications to semilinear elliptic PDE,” The Annals of Probability, vol. 25, no. 3, pp. 1135–1159, 1997.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNetN. El Karoui, S. Peng, and M. C. Quenez, “Backward stochastic differential equations in finance,” Mathematical Finance, vol. 7, no. 1, pp. 1–71, 1997.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNetN. El-Karoui and S. Hamadène, “BSDEs and risk-sensitive control, zero-sum and nonzero-sum game problems of stochastic functional differential equations,” Stochastic Processes and Their Applications, vol. 107, no. 1, pp. 145–169, 2003.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNetS. Hamadène and J.-P. Lepeltier, “Zero-sum stochastic differential games and backward equations,” Systems & Control Letters, vol. 24, no. 4, pp. 259–263, 1995.
View at: Google Scholar | Zentralblatt MATH | MathSciNetM. Kobylanski, “Résultats d'existence et d'unicité pour des équations différentielles stochastiques rétrogrades avec des générateurs à croissance quadratique,” Comptes Rendus de l'Académie des Sciences. Série I. Mathématique, vol. 324, no. 1, pp. 81–86, 1997.
View at: Google Scholar | Zentralblatt MATH | MathSciNetJ. P. Lepeltier and J. San Martin, “Backward stochastic differential equations with continuous coefficient,” Statistics & Probability Letters, vol. 32, no. 4, pp. 425–430, 1997.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNetA. Løkka, “Martingale representation, chaos expansion and Clark-Ocone formulas,” MPS-RR 1999-22, 1999.
View at: Google ScholarJ.-L. Menaldi and M. Robin, “Reflected diffusion processes with jumps,” The Annals of Probability, vol. 13, no. 2, pp. 319–341, 1985.
View at: Google Scholar | Zentralblatt MATH | MathSciNetD. Nualart and W. Schoutens, “Chaotic and predictable representations for Lévy processes,” Stochastic Processes and Their Applications, vol. 90, no. 1, pp. 109–122, 2000.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNetD. Nualart and W. Schoutens, “Backward stochastic differential equations and Feynman-Kac formula for Lévy processes, with applications in finance,” Bernoulli, vol. 7, no. 5, pp. 761–776, 2001.
View at: Google Scholar | Zentralblatt MATH | MathSciNetE. Pardoux, “Backward stochastic differential equations and viscosity solutions of systems of semilinear parabolic and elliptic PDEs of second order,” in Stochastic Analysis and Related Topics, VI (Geilo, 1996), vol. 42 of Progr. Probab., pp. 79–127, Birkhäuser Boston, Massachusetts, 1998.
View at: Google Scholar | Zentralblatt MATH | MathSciNetE. Pardoux, “BSDEs, weak convergence and homogenization of semilinear PDEs,” in Nonlinear Analysis, Differential Equations and Control (Montreal, QC, 1998), vol. 528 of NATO Sci. Ser. C Math. Phys. Sci., pp. 503–549, Kluwer Academic, Dordrecht, 1999.
View at: Google Scholar | Zentralblatt MATH | MathSciNetE. Pardoux and S. G. Peng, “Adapted solution of a backward stochastic differential equation,” Systems & Control Letters, vol. 14, no. 1, pp. 55–61, 1990.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNetE. Pardoux and S. Zhang, “Generalized BSDEs and nonlinear Neumann boundary value problems,” Probability Theory and Related Fields, vol. 110, no. 4, pp. 535–558, 1998.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNetS. G. Peng, “Probabilistic interpretation for systems of quasilinear parabolic partial differential equations,” Stochastics and Stochastics Reports, vol. 37, no. 1-2, pp. 61–74, 1991.
View at: Google Scholar | Zentralblatt MATH | MathSciNetP. E. Protter, Stochastic Integration and Differential Equations, Stochastic Modeling and Applied Probability, Springer, Berlin, 2nd edition, 2005, Version 2.1.