Skip to main content
SpringerLink
Log in

BSDEs with jumps and path-dependent parabolic integro-differential equations

  • Published:
Chinese Annals of Mathematics, Series B Aims and scope Submit manuscript

Abstract

This paper deals with backward stochastic differential equations with jumps, whose data (the terminal condition and coefficient) are given functions of jump-diffusion process paths. The author introduces a type of nonlinear path-dependent parabolic integrodifferential equations, and then obtains a new type of nonlinear Feynman-Kac formula related to such BSDEs with jumps under some regularity conditions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Barles, G., Buckdahn, R. and Pardoux, E., Backward stochastic differential equations and integral-partial differential equations, Stochastics Stochastics Rep., 60, 1997, 57–83.

    Article  MATH  MathSciNet  Google Scholar 

  2. Bismut, J. M., Conjugate convex functions in optimal stochastic control, J. Math. Anal. Appl., 44, 1973, 384–404.

    Article  MathSciNet  Google Scholar 

  3. Buckdahn, R. and Pardoux E., BSDEs with jumps and associated integro-partial differential equations, preprint.

  4. Cont, R. and Fournié, D. A., Change of variable formulas for non-anticipative functionals on path space, J. Funct. Anal., 259(4), 2010, 1043–1072.

    Article  MATH  MathSciNet  Google Scholar 

  5. Cont, R. and Fournié, D. A., A functional extension of the Itô formula, C. R. Math. Acad. Sci. Paris., 348(1), 2010, 57–61.

    Article  MATH  MathSciNet  Google Scholar 

  6. Cont, R. and Fournie, D. A., Functional Ito calculus and stochastic integral representation of martingales, Ann. Probab., 41(1), 2013, 109–133.

    Article  MATH  MathSciNet  Google Scholar 

  7. Dupire, B., Functional Itô calculus, Portfolio Research Paper 2009-04, Bloomberg.

  8. Ekren, I., Keller, C., Touzi, N. and Zhang, J., On viscosity solutions of path dependent PDEs, Ann. Probab., 42(1), 2014, 204–236.

    Article  MATH  MathSciNet  Google Scholar 

  9. Ekren, I., Touzi, N. and Zhang, J., Viscosity solutions of fully nonlinear parabolic path dependent PDEs: Part I. arXiv: 1210.0006

  10. Ekren, I., Touzi, N. and Zhang, J., Viscosity solutions of fully nonlinear parabolic path dependent PDEs: Part II. arXiv: 1210.0007

  11. El Karoui, N., Peng, S. and Quenez, M. C., Backward stochastic differential equation in finance, Math. Finance, 7(1), 1997, 1–71.

    Article  MATH  MathSciNet  Google Scholar 

  12. Hu, Y. and Ma, J., Nonlinear Feynman-Kac formula and discrete-functional-type BSDEs with continuous coefficients, Stochastic Process. Appl., 112(1), 2004, 23–51.

    Article  MATH  MathSciNet  Google Scholar 

  13. Ma, J. and Yong, J., Forward-backward stochastic differential equations and their applications, Lecture Notes in Mathematics, 1702, Springer-Verlag, Berlin, 1999.

  14. Ma, J. and Zhang, J., Representation theorems for backward SDEs, Ann. Appl. Probab., 12(4), 2002, 1390–1418.

    Article  MATH  MathSciNet  Google Scholar 

  15. Ma, J. and Zhang, J., Path regularity for solutions of backward stochastic differential equations, Probab. Theory Related Fields., 122(2), 2002, 163–190.

    Article  MATH  MathSciNet  Google Scholar 

  16. Pardoux, E. and Peng, S., Adapted solutions of backward stochastic equations, Systems Control Lett., 14, 1990, 55–61.

    Article  MATH  MathSciNet  Google Scholar 

  17. Pardoux, E. and Peng, S., Backward stochastic differential equations and quasilinear parabolic partial differential equations, Stochastic Partial Differential Equations and Their Applications, B. L. Rozuvskii and R. B. Sowers (eds.), Lect. Notes Control Inf. Sci., Vol. 176, Springer-Verlag, Berlin, Heidelberg, New York, 1992, 200–217.

    Article  MathSciNet  Google Scholar 

  18. Peng, S., Probabilistic interpretation for systems of quasilinear parabolic partial differential equation, Stochastics Stochastics Rep., 37, 1991, 61–74.

    Article  MATH  MathSciNet  Google Scholar 

  19. Peng, S., A nonlinear Feynman-Kac formula and applications, Control Theory, Stochastic Analysis and Applications (Hangzhou, 1991), World Sci. Publ., River Edge, NJ, 1992, 173–184.

    Google Scholar 

  20. Peng, S., Note on viscosity solution of path-dependent PDE and G-martingales. arXiv: 1106.1144

  21. Peng, S. and Wang, F., BSDE, path-dependent PDE and nonlinear Feynman-Kac Formula. arXiv: 1108.4317

  22. Tang, S. and Li, X., Necessary conditions for optimal control of stochastic systems with random jumps, SIAM J. Control Optim., 32(5), 1994, 1447–1475.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute for Advanced Research and School of Mathematics, Shandong University, Jinan, 250100, China

    Falei Wang

Authors
  1. Falei Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Falei Wang.

Additional information

This work was supported by the National Natural Science Foundation of China (Nos. 10921101, 11471190), the Shandong Provincial Natural Science Foundation of China (No. ZR2014AM002) and the Programme of Introducing Talents of Discipline to Universities of China (No. B12023).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, F. BSDEs with jumps and path-dependent parabolic integro-differential equations. Chin. Ann. Math. Ser. B 36, 625–644 (2015). https://doi.org/10.1007/s11401-015-0917-5

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11401-015-0917-5

Keywords

2000 MR Subject Classification

Search

Navigation