On Solutions of Backward Stochastic Differential Equations with Jumps and Stochastic Control

  • Situ Rong

Abstract

We relax conditions on coefficients given in [7] for the existence of solu-tions to backward stochastic differential equations (BSDE) with jumps. Counter examples are given to show that such conditions can not be weakened further in some sense. The existence of a solution for some continuous BSDE with coefficients b(t, y, q) having a quadratic growth in q, having a greater than linear growth in y, and are unbounded in y belonging to a finite interval, is also obtained. Then we obtain an existence and uniqueness result for the Sobolev solution to some integro-differential equation (IDE) under weaker conditions. Some Markov properties for solutions to BSDEs associated with some forward SDEs are also discussed and a Feynman-Kac formula is also obtained. Finally, we obtain probably the first results on the existence of non-Lipschitzian optimal controls for some special stochastic control problems with respect to such BSDE systems with jumps, where some optimal control problem is also explained in the financial market.

Keywords

Dition 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Bardhan, I. and Chao, X. (1993). Pricing options on securities with discontinuous returns. Stochastic Process. Appl.48, 123–137.MathSciNetMATHCrossRefGoogle Scholar
  2. [2]
    Bensoussan, A. and J.L. Lions. (1984). Impulse Control and Quasi-Variational Inequalities. Gautheir-Villars.Google Scholar
  3. [3]
    El Karoui, N., Peng, S., and Quenez, M.C. (1997). Backward stochastic differential equations in finance. Math. Finance7, 1–71.MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    Ladyzenskaja, O. A., Solonnikov, V. A. and Uralceva, N. N. (1968). Linear and Quasilinear Equations of Parabolic Type. Translation of Monographs 23, AMS Providence, Rode Island.Google Scholar
  5. [5]
    Peng, S. (1993). Backward stochastic differential equations and applications to optimal control. Appl. Math.Optim.27, 125–144.MATHCrossRefGoogle Scholar
  6. [6]
    Situ Rong (1996). On comparison theorem of solutions to backward stochastic differential equations with jumps and its applications. Proc. of the 3rd CSIAM Conf. on Systems and Control, LiaoNing, China, 46–50.Google Scholar
  7. [7]
    Situ Rong. (1997). On solutions of backward stochastic differential equations with jumps and applications. Stochastic Process. Appl.66, 209–236.MathSciNetMATHCrossRefGoogle Scholar
  8. [8]
    Situ Rong. (1999). Comparison theorem of solutions to BSDE with jumps and viscosity solution to a generalized Hamilton-JacobiBellman equation. In Control of Distributed Parameter and Stochastic Systems, eds: S. Chen, X. Li, J. Yong, X.Y. Zhou, Kluwer Acad. Pub., Boston, 275–282.Google Scholar
  9. [9]
    Situ Rong (1999). On comparison theorems and existence of solutions to backward stochastic differential equations with jumps and with discontinuous coefficients. The 26th Conference on Stochastic Processes and their Applications, 14–18, June, Beijing.Google Scholar
  10. [10]
    Situ Rong (1999). Reflecting Stochastic Differential Equations with Jumps and Applications. CRC Research Notes in Mathematics 408, Chapman Sc Hall/CRC.Google Scholar

Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

  • Situ Rong
    • 1
  1. 1.Department of MathematicsZhongshan UniversityGuangzhouChina

Personalised recommendations