Existence of a weak solution to a Markovian BSDE with discontinuous coefficients

References

[1] K. Bahlali, Flows of homeomorphisms of stochastic differential equations with measurable drift, Stochastics Stochastics Rep. 67 (1999), no. 1–2, 53–82. 10.1080/17442509908834203Search in Google Scholar

[2] K. Bahlali, Backward stochastic differential equations with locally Lipschitz coefficient, C. R. Acad. Sci. Paris Sér. I Math. 333 (2001), no. 5, 481–486. 10.1016/S0764-4442(01)02063-8Search in Google Scholar

[3] K. Bahlali, Existence and uniqueness of solutions for BSDEs with locally Lipschitz coefficient, Electron. Commun. Probab. 7 (2002), 169–179. 10.1214/ECP.v7-1058Search in Google Scholar

[4] K. Bahlali, A. Elouaflin and E. Pardoux, Homogenization of semilinear PDEs with discontinuous averaged coefficients, Electron. J. Probab. 14 (2009), 477–499. 10.1214/EJP.v14-627Search in Google Scholar

[5] K. Bahlali, E. Essaky and M. Hassani, Existence and uniqueness of multidimensional BSDEs and of systems of degenerate PDEs with superlinear growth generator, SIAM J. Math. Anal. 47 (2015), no. 6, 4251–4288. 10.1137/130947933Search in Google Scholar

[6] K. Bahlali, E. H. Essaky, M. Hassani and E. Pardoux, Existence, uniqueness and stability of backward stochastic differential equations with locally monotone coefficient, C. R. Math. Acad. Sci. Paris 335 (2002), no. 9, 757–762. 10.1016/S1631-073X(02)02542-6Search in Google Scholar

[7] K. Bahlali, B. Mezerdi, M. N’zi and Y. Ouknine, Weak solutions and a Yamada–Watanabe theorem for FBSDEs, Random Oper. Stoch. Equ. 15 (2007), no. 3, 271–285. 10.1515/rose.2007.016Search in Google Scholar

[8] J.-M. Bismut, Conjugate convex functions in optimal stochastic control, J. Math. Anal. Appl. 44 (1973), 384–404. 10.1016/0022-247X(73)90066-8Search in Google Scholar

[9] R. Buckdahn, H.-J. Engelbert and A. Răşcanu, On weak solutions of backward stochastic differential equations, Theory Probab. Appl. 49 (2005), no. 1, 70–108. 10.4213/tvp237Search in Google Scholar

[10] L. Caffarelli, M. G. Crandall, M. Kocan and A. Swiech, On viscosity solutions of fully nonlinear equations with measurable ingredients, Comm. Pure Appl. Math. 49 (1996), no. 4, 365–397. 10.1002/(SICI)1097-0312(199604)49:4<365::AID-CPA3>3.3.CO;2-VSearch in Google Scholar

[11] F. Delarue and G. Guatteri, Weak existence and uniqueness for forward-backward SDEs, Stochastic Process. Appl. 116 (2006), no. 12, 1712–1742. 10.1016/j.spa.2006.05.002Search in Google Scholar

[12] N. El Karoui, Backward stochastic differential equations: A general introduction, Backward Stochastic Differential Equations, Pitman Res. Notes Math. Ser. 364, Longman, Harlow (1997), 7–26. 10.1111/1467-9965.00022Search in Google Scholar

[13] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd ed., Grundlehren Math. Wiss. 224, Springer, Berlin, 1983. Search in Google Scholar

[14] S. Hamadene, Équations différentielles stochastiques rétrogrades: les cas localement lipschitzien, Ann. Inst. H. Poincaré Probab. Statist. 32 (1996), no. 5, 645–659. Search in Google Scholar

[15] S. Hamadene, J.-P. Lepeltier and S. Peng, BSDEs with continuous coefficients and stochastic differential games, Backward Stochastic Differential Equations, Pitman Res. Notes Math. Ser. 364, Longman, Harlow (1997), 115–128. Search in Google Scholar

[16] R. Khasminskii and N. Krylov, On averaging principle for diffusion processes with null-recurrent fast component, Stochastic Process. Appl. 93 (2001), no. 2, 229–240. 10.1016/S0304-4149(00)00097-1Search in Google Scholar

[17] D. Kim and N. V. Krylov, Parabolic equations with measurable coefficients, Potential Anal. 26 (2007), no. 4, 345–361. 10.1007/s11118-007-9042-8Search in Google Scholar

[18] N. V. Krylov, On weak uniqueness for some diffusions with discontinuous coefficients, Stochastic Process. Appl. 113 (2004), no. 1, 37–64. 10.1016/j.spa.2004.03.012Search in Google Scholar

[19] N. V. Krylov, Lectures on Elliptic and Parabolic Equations in Sobolev Spaces, Grad. Stud. Math. 96, American Mathematical Society, Providence, 2008. 10.1090/gsm/096Search in Google Scholar

[20] O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Ural’tseva, Linaer and Quasi-Linear Equations of Parabolic Type, American Mathematical Society, Providence, 1968. Search in Google Scholar

[21] J. P. Lepeltier and J. San Martin, Backward stochastic differential equations with continuous coefficient, Statist. Probab. Lett. 32 (1997), no. 4, 425–430. 10.1016/S0167-7152(96)00103-4Search in Google Scholar

[22] J.-P. Lepeltier and J. San Martín, Existence for BSDE with superlinear-quadratic coefficient, Stochastics Stochastics Rep. 63 (1998), no. 3–4, 227–240. 10.1080/17442509808834149Search in Google Scholar

[23] S. Nakao, On the pathwise uniqueness of solutions of one-dimensional stochastic differential equations, Osaka Math. J. 9 (1972), 513–518. Search in Google Scholar

[24] E. Pardoux, Homogenization of linear and semilinear second order parabolic PDEs with periodic coefficients: A probabilistic approach, J. Funct. Anal. 167 (1999), no. 2, 498–520. 10.1006/jfan.1999.3441Search in Google Scholar

[25] E. Pardoux and S. G. Peng, Adapted solution of a backward stochastic differential equation, Systems Control Lett. 14 (1990), no. 1, 55–61. 10.1016/0167-6911(90)90082-6Search in Google Scholar

[26] J. Simon, Compacts sets in the espace L p ⁢ ( 0 , T ; B ) , Ann. Mat. Pura Appl. (4) 146 (1987), 65–96. 10.1007/BF01762360Search in Google Scholar

[27] A. V. Skorokhod, Studies in the Theory of Random Processes, Addison-Wesley, Reading, 1965. Search in Google Scholar

[28] D. W. Stroock and S. R. S. Varadhan, Multidimensional Diffusion Processes, Grundlehren Math. Wiss. 233, Springer, Berlin, 1979. Search in Google Scholar