Skip to main content
SpringerLink
Log in

BSDE, path-dependent PDE and nonlinear Feynman-Kac formula

  • Articles
  • Progress of Projects Supported by NSFC
  • Published:
Science China Mathematics Aims and scope Submit manuscript

Abstract

We introduce a new type of path-dependent quasi-linear parabolic PDEs in which the continuous paths on an interval [0, t] become the basic variables in the place of classical variables (t, x) ∈ [0, T] × Rd. This new type of PDEs are formulated through a classical BSDE in which the terminal values and the generators are allowed to be general function of Brownian motion paths. In this way, we establish the nonlinear Feynman-Kac formula for a general non-Markovian BSDE. Some main properties of solutions of this new PDEs are also obtained.

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

    Article  MathSciNet  Google Scholar 

  2. Bouchard B, Touzi N. Discrete time approximation and Monte Carlo simulation of backward stochastic differential equations. Stochastic Process Appl, 2004, 111: 175–206

    Article  MathSciNet  MATH  Google Scholar 

  3. Briand P, Delyon B, Hu Y, et al. L p-solutions of backward stochastic differential equations. Stochastic Process Appl, 2003, 108: 109–129

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  6. Cont R, Fournié D-A. Functional Itô calculus and stochastic integral representation of martingales. Ann Probab, 2013, 41: 109–133

    Article  MathSciNet  MATH  Google Scholar 

  7. Douglas J, Ma J, Protter P. Numerical methods for forward-backward stochastic differential equations. Ann Appl Probab, 1996, 6: 940–968

    Article  MathSciNet  MATH  Google Scholar 

  8. Dupire B. Functional Itô calculus. Bloomberg Portfolio Research Paper No. 2009-04-FRONTIERS. Http://ssrn.com/abstract=1435551

  9. Ekren I, Keller C, Touzi N, et al. On viscosity solutions of path dependent PDEs. Ann Probab, 2014, 42: 204–236

    Article  MathSciNet  MATH  Google Scholar 

  10. Ekren I, Touzi N, Zhang J F. Viscosity solutions of fully nonlinear parabolic path dependent PDEs: Part I. ArXiv:1210.0006, 2012

    Google Scholar 

  11. Ekren I, Touzi N, Zhang J F. Viscosity solutions of fully nonlinear parabolic path dependent PDEs: Part II. ArXiv:1210.0007, 2012

    Google Scholar 

  12. El Karoui N, Peng S G, Quenez M C. Backward stochastic differential equation in finance. Math Finance, 1997, 7: 1–71

    Article  MathSciNet  MATH  Google Scholar 

  13. Gobet E, Lemor J P, Warin X. A regression-based Monte Carlo method to solve backward stochastic differential equations. Ann Appl Probab, 2005, 15: 2172–2202

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  15. Ma J, Zhang J F. Representation theorems for backward stochastic differential equations. Ann Appl Probab, 2002, 12: 1390–1418

    Article  MathSciNet  MATH  Google Scholar 

  16. Ma J, Zhang J F. Path regularity for solutions of backward stochastic differential equations. Probab Theory Related Fields, 2002, 122: 163–190

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  20. Peng S G. A nonlinear Feynman-Kac formula and applications. In: Control Theory, Stochastic Analysis and Applications. River Edge: World Sci Publ, 1991, 173–184

    Google Scholar 

  21. Peng S G. Nonlinear expectations and nonlinear Markov chain. Chin Ann Math B, 2005, 26: 159–184

    Article  MATH  Google Scholar 

  22. Peng S G. G-expectation, G-Brownian Motion and Related Stochastic Calculus of Itô type. In: Stochastic Analysis and Applications. Abel Symp, vol. 2. Berlin: Springer, 2007, 541–567

    Article  Google Scholar 

  23. Peng S G. Backward stochastic differential equation, nonlinear expectation and their applications. In: Proceedings of the International Congress of Mathematicians, vol. 1. New Delhi: Hindustan Book Agency, 2011, 393–432

    Google Scholar 

  24. Peng S G. Note on viscosity solution of path-dependent PDE and G-martingales lecture notes. ArXiv:1106.1144v2, 2011

    Google Scholar 

  25. Peng S G, Song Y S. G-expectation weighted Sobolev spaces, backward SDE and path dependent PDE. ArXiv:1305.4722, 2013

    Google Scholar 

  26. Peng S G, Xu M Y. Numerical algorithms for backward stochastic differential equations with 1-dimenisonal Brownian motion: Convergence and simulations. ArXiv:0611864, 2006

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics, Shandong University, Jinan, 250100, China

    ShiGe Peng & FaLei Wang

  2. Qilu Institute of Finance, Shandong University, Jinan, 250100, China

    ShiGe Peng & FaLei Wang

  3. Institute for Advanced Research, Shandong University, Jinan, 250100, China

    FaLei Wang

Authors
  1. ShiGe Peng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. FaLei Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to ShiGe Peng.

Additional information

Citation: Peng S G, Wang F L. BSDE, path-dependent PDE and nonlinear Feynman-Kac formula. Sci China Math, 2016, 59, doi: 10.1007/s11425-015-5086-1

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Peng, S., Wang, F. BSDE, path-dependent PDE and nonlinear Feynman-Kac formula. Sci. China Math. 59, 19–36 (2016). https://doi.org/10.1007/s11425-015-5086-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11425-015-5086-1

Keywords

MSC(2010)

Search

Navigation