Advertisement

Probability Theory and Related Fields

, Volume 98, Issue 3, pp 339–359 | Cite as

Solving forward-backward stochastic differential equations explicitly — a four step scheme

  • Jin Ma
  • Philip Protter
  • Jiongmin Yong
Article

Summary

In this paper we investigate the nature of the adapted solutions to a class of forward-backward stochastic differential equations (SDEs for short) in which the forward equation is non-degenerate. We prove that in this case the adapted solution can always be sought in an “ordinary” sense over an arbitrarily prescribed time duration, via a direct “Four Step Scheme”. Using this scheme, we further prove that the backward components of the adapted solution are determined explicitly by the forward components via the solution of a certain quasilinear parabolic PDE system. Moreover the uniqueness of the adapted solutions (over an arbitrary time duration), as well as the continuous dependence of the solutions on the parameters, can all be proved within this unified framework. Some special cases are studied separately. In particular, we derive a new form of the integral representation of the Clark-Haussmann-Ocone type for functionals (or functions) of diffusions, in which the conditional expectation is no longer needed.

Mathematics Subject Classification (1991)

60H10 60H20 60G44 35K55 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Antonelli, F.: Backward-forward stochastic differential equations. Ann. Appl. Probab.3, 777–793 (1993)Google Scholar
  2. 2.
    Deimling, K.: Nonlinear functional analysis. Berlin Heidelberg Springer New York: 1988Google Scholar
  3. 3.
    Haussmann, U. G.: On the integral representation of functionals of Itô's processes. Stochastics3, 17–27 (1979)Google Scholar
  4. 4.
    Krylov, N.V.: Controlled diffusion processes. Berlin Heidelberg New York: Springer 1980Google Scholar
  5. 5.
    Ladyzenskaja, O. A., Solonnikov, V. A. Ural'ceva, N. N.: Linear and quasilinear equations of parabolic type, Providence, RI: Am. Math. Soc. 1968Google Scholar
  6. 6.
    Ma, J., Yong, J.: Solvability of forward-backwards SDEs and the nodal set of Hamilton-Jacobi-equations. (Preprint 1993)Google Scholar
  7. 7.
    Nualart, D.: Noncausal stochastic integrals and calculus. In: Korezlioglu, H. Ustunel, A.S. (eds.) Stochastic analysis and related topics. (Lect. Notes Math., vol. 1316, pp. 80–129) Berlin Heidelberg New York: Springer 1986Google Scholar
  8. 8.
    Ocone, D.: Malliavin's calculus and stochastic integral representations of functionals of diffusion processes. Stochastics12, 161–185 (1984)Google Scholar
  9. 9.
    Pardoux, E., Peng, S.: Adapted solution of a backward stochastic differential equation. Syst. Control Lett.14, 55–61 (1990)Google Scholar
  10. 10.
    Pardoux, E., Peng, S.: Backward stochastic differential equations and quasi-linear parabolic partial differential equations. In: Rozovskii, B. L., Sowers, R. S. (eds.) Stochastic partial differential equations and their applications (Lect. Notes Control. Inf. Sci.176, pp. 200–217) Berlin Heidelberg New York: Springer 1992Google Scholar
  11. 11.
    Peng, S.: Adapted solution of backward stochastic equation and related partial differential equations. (Preprint 1993)Google Scholar
  12. 12.
    Protter, P.: Stochastic integration and differential equations, a new approach. Berlin Heidelberg New York: Springer 1990Google Scholar
  13. 13.
    Wiegner, M.: Global solutions to a class of strongly coupled parabolic systems. Math. Ann.292, 711–727 (1992)Google Scholar
  14. 14.
    Zhou, X. Y.: A unified treatment of maximum principle and dynamic programming in stochastic controls. Stochastics Stochastics Rep.36, 137–161 (1991)Google Scholar

Copyright information

© Springer-Verlag 1994

Authors and Affiliations

  • Jin Ma
    • 1
  • Philip Protter
    • 1
  • Jiongmin Yong
    • 2
  1. 1.Department of MathematicsPurdue UniversityWest LafayetteUSA
  2. 2.Department of MathematicsFudan UniversityShanghaiPeople's Republic of China

Personalised recommendations