Dans cette Note on démontre que si g est continue, monotone, de croissance quelconque en y, g uniformément continue en z et est de carré intégrable, alors pour toute condition finale ξ de carré intégrable, en dimension un, l'équation différentielle stochastique rétrograde (BSDE) de générateur g, a une solution unique. Ce résultat généralise des résultats connus dans le cas de la dimension un.
In this Note, we prove that if g is continuous, monotonic and has a general growth in y, g is uniformly continuous in z, and is square integrable, then for each square integrable terminal condition ξ, the one-dimensional backward stochastic differential equation (BSDE) with the generator g has a unique solution. This generalizes some corresponding (one-dimensional) results.
@article{CRMATH_2010__348_1-2_89_0, author = {Sheng-Jun Fan and Long Jiang}, title = {Uniqueness result for the {BSDE} whose generator is monotonic in \protect\emph{y} and uniformly continuous in \protect\emph{z}}, journal = {Comptes Rendus. Math\'ematique}, pages = {89--92}, publisher = {Elsevier}, volume = {348}, number = {1-2}, year = {2010}, doi = {10.1016/j.crma.2009.10.023}, language = {en}, }
TY - JOUR AU - Sheng-Jun Fan AU - Long Jiang TI - Uniqueness result for the BSDE whose generator is monotonic in y and uniformly continuous in z JO - Comptes Rendus. Mathématique PY - 2010 SP - 89 EP - 92 VL - 348 IS - 1-2 PB - Elsevier DO - 10.1016/j.crma.2009.10.023 LA - en ID - CRMATH_2010__348_1-2_89_0 ER -
Sheng-Jun Fan; Long Jiang. Uniqueness result for the BSDE whose generator is monotonic in y and uniformly continuous in z. Comptes Rendus. Mathématique, Volume 348 (2010) no. 1-2, pp. 89-92. doi : 10.1016/j.crma.2009.10.023. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.10.023/
[1] Backward stochastic differential equations with locally Lipschitz coefficient, C. R. Acad. Sci. Paris, Ser. I, Volume 331 (2001), pp. 481-486
[2] solutions of backward stochastic differential equations, Stochastic Processes and Their Applications, Volume 108 (2003), pp. 109-129
[3] Quadratic BSDEs with convex generators and unbounded terminal conditions, Probability Theory and Related Fields, Volume 141 (2008), pp. 543-567
[4] One-dimensional BSDE's whose coefficient is monotonic in y and non-Lipschitz in z, Bernoulli, Volume 13 (2007) no. 1, pp. 80-91
[5] Viscosity solutions — a primer (I. Capuzzo Dolcetta; P.L. Lions, eds.), Viscosity Solutions and Applications, Lecture Notes in Mathematics, vol. 1660, Springer, Berlin, 1997, pp. 1-43
[6] Semimartingal Theory and Stochastic Calculus, Science Press and CRC Press, Inc., 1992
[7] A uniqueness theorem for the solution of backward stochastic differential equations, C. R. Acad. Sci. Paris, Ser. I, Volume 346 (2008), pp. 439-444
[8] Backward stochastic differential equations and partial equations with quadratic growth, Ann. Probab., Volume 28 (2000), pp. 259-276
[9] Backward stochastic differential equations with continuous coefficient, Statistics and Probability Letters, Volume 32 (1997), pp. 425-430
[10] Adapted solutions of backward stochastic differential equations with non-Lipschitz coefficients, Stochastic Process and Their Applications, Volume 58 (1995), pp. 281-292
[11] Adapted solution of a backward stochastic differential equation, Systems Control Letters, Volume 14 (1990), pp. 55-61
[12] BSDEs, weak convergence and homogenization of semilinear PDEs, Montreal, QC, 1998, Kluwer Academic Publishers, Dordrecht (1999), pp. 503-549
[13] Monotonic limit theorem of BSDE and nonlinear decomposition theorem of Doob–Meyer's type, Probability Theory and Related Fields, Volume 113 (1999), pp. 473-499
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☆ Supported by the National Natural Science Foundation of China (No. 10671205), National Basic Research Program of China (No. 2007CB814901), Youth Foundation of China University of Mining & Technology (Nos. 2006A041 and 2007A029) and Qing Lan Project.
Commentaires - Politique
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Some results on the uniqueness of generators of backward stochastic differential equations
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Representation theorems for generators of backward stochastic differential equations
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