Representation Theorem for Generators of BSDEs with Monotonic and Polynomial-Growth Generators in the Space of Processes

ShengJun Fan (China University of Mining and Technology)
Long Jiang (China University of Mining and Technology)
YingYing Xu (China University of Mining and Technology)

Abstract


In this paper, on the basis of some recent works of Fan, Jiang and Jia, we establish a representation theorem in the space of processes for generators of BSDEs with monotonic and polynomial-growth generators, which generalizes the corresponding results in Fan (2006, 2007), and Fan and Hu (2008).

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 830-844

Publication Date: April 22, 2011

DOI: 10.1214/EJP.v16-873

References

  1. Ph. Briand, F. Coquet, Y. Hu, J. Memin and S.G. Peng. A converse comparison theorem for BSDEs and related properties of g-expectation. Electron. Comm. Probab. 5 (2000), 101--117 (electronic). MR1781845 (2001h:60105)
  2. Ph. Briand, B. Delyon, Y. Hu, E. Pardoux and L. Stoica. Lp solutions of backward stochastic differential equations. Stochastic Process. Appl. 108 (2003), no. 1, 109--129. MR2008603 (2005c:60073)
  3. Ph. Briand, J.P. Lepeltier and J. San Martin. One-dimensional backward stochastic differential equations whose coefficient is monotonic in y and non-Lipschitz in z. Bernoulli 13 (2007), no. 1, 80--91. MR2307395 (2007k:60190)
  4. Z.J. Chen, R. Kulperger and L. Jiang. Jensen's inequality for g-expectation. I. C. R. Math. Acad. Sci. Paris 337 (2003), no. 11, 725--730. MR2030410 (2004k:60162)
  5. S.J. Fan. Jensen's inequality for g-expectation on convex (concave) function. (Chinese) Chinese Ann. Math. Ser. A 27 (2006), no. 5, 635--644; translation in Chinese J. Contemp. Math. 27 (2006), no. 4, 401--412 (2007) MR2266069 (2008e:60041)
  6. S.J. Fan. A relationship between the conditional g-evaluation system and the generator g and its applications. Acta Math. Sin. (Engl. Ser.) 23 (2007), no. 8, 1427--1434. MR2320749 (2008g:60179)
  7. S.J. Fan and J.H. Hu. A limit theorem for solutions to BSDEs in the space of processes. Statist. Probab. Lett. 78 (2008), no. 8, 1024--1033. MR2418920 (2009f:60071)
  8. S.J. Fan and L. Jiang. Existence and uniqueness result for a backward stochastic differential equation whose generator is Lipschitz continuous in y and uniformly continuous in z. Journal of Applied Mathematics and Computing, 2010. DOI:10.1007/s12190-010-0384-9.
  9. S.J. Fan and L. Jiang. A representation theorem for generators of BSDEs with continuous linear-growth generators in the space of processes. J. Comput. Appl. Math. 235 (2010), no. 3, 686--695. MR2719808
  10. G. Jia. Backward stochastic differential equations, g-expectations and related semilinear PDEs. PH.D Thesis, ShanDong University, China, 2008.
  11. L. Jiang and Z.J. Chen. On Jensen's inequality for g-expectation. Chinese Ann. Math. Ser. B 25 (2004), no. 3, 401--412. MR2086132 (2005h:60168)
  12. L. Jiang. Some results on the uniqueness of generators of backward stochastic differential equations. C. R. Math. Acad. Sci. Paris 338 (2004), no. 7, 575--580. MR2057033 (2005e:60141)
  13. L. Jiang. Converse comparison theorems for backward stochastic differential equations. Statist. Probab. Lett. 71 (2005), no. 2, 173--183. MR2126773 (2005j:60116)
  14. L. Jiang. Representation theorems for generators of backward stochastic differential equations. C. R. Math. Acad. Sci. Paris 340 (2005), no. 2, 161--166. MR2116776 (2005j:60125)
  15. L. Jiang. Representation theorems for generators of backward stochastic differential equations and their applications. Stochastic Process. Appl. 115 (2005), no. 12, 1883--1903. MR2178500 (2006g:60084)
  16. L. Jiang. Nonlinear expectation--g-expectation theory and its applications in finance. PH.D Thesis, ShanDong University, China, 2005.
  17. L. Jiang. Limit theorem and uniqueness theorem of backward stochastic differential equations. Sci. China Ser. A 49 (2006), no. 10, 1353--1362. MR2287264 (2008c:60053)
  18. L. Jiang. Convexity, translation invariance and subadditivity for g-expectations and related risk measures. Ann. Appl. Probab. 18 (2008), no. 1, 245--258. MR2380898 (2009f:60079)
  19. J.P. Lepeltier and J. San Martin. Backward stochastic differential equations with continuous coefficient. Statist. Probab. Lett. 32 (1997), no. 4, 425--430. MR1602231 (98j:60082)
  20. Y.C. Liu, L. Jiang and Y.Y. Xu. A local limit theorem for solutions of BSDEs with Mao's non-Lipschitz generator. Acta Math. Appl. Sin. Engl. Ser. 24 (2008), no. 2, 329--336. MR2411506(2009f:60072)
  21. X.R. Mao. Adapted solutions of backward stochastic differential equations with non-Lipschitz coefficients. Stochastic Process. Appl. 58 (1995), no. 2, 281--292. MR1348379 (96f:60099)
  22. E. Pardoux and S.G. Peng. Adapted solution of a backward stochastic differential equation. Systems Control Lett. 14 (1990), no. 1, 55--61. MR1037747 (91e:60171)
Bookmark and Share


Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.