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Feynman-Kac Formulas, Backward Stochastic Differential Equations and Markov Processes

  • Jan A. Van Casteren

Abstract

In this paper we explain the notion of stochastic backward differential equations and its relationship with classical (backward) parabolic differential equations of second order. The paper contains a mixture of stochastic processes like Markov processes and martingale theory and semi-linear partial differential equations of parabolic type. Some emphasis is put on the fact that the whole theory generalizes Feynman-Kac formulas. A new method of proof of the existence of solutions is given. All the existence arguments are based on rather precise quantitative estimates.

Keywords

Brownian Motion Markov Process Monotonicity Condition Unique Pair Malliavin Calculus 
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References

  1. [1]
    D. Bakry, L’hypercontractivité et son utilisation en théorie de semigroupes, Lecture Notes in Math., vol. 1581, pp. 1–114, Springer Verlag, Berlin, 1994, P. Bernard (editor).Google Scholar
  2. [2]
    _____, Functional inequalities for Markov semigroups, Probability measures on groups: recent directions and trends, Tata Inst. Fund. Res., Mumbai, 2006, pp. 91–147.Google Scholar
  3. [3]
    D. Bakry and M. Ledoux, A logarithmic Sobolev form of the Li-Yau parabolic inequality, Revista Mat. Iberoamericana 22 (2006), 683–702, to appear.MATHMathSciNetGoogle Scholar
  4. [4]
    V. Bally, É Pardoux, and L. Stoica, Backward stochastic differential equations associated to a symmetric Markov process, Potential Analysis 22 (2005), no. 1, 17–60.MATHMathSciNetGoogle Scholar
  5. [5]
    M.T. Barlow, R.F. Bass, and T. Kumagai, Note on the equivalence of parabolic Harnack inequalities and heat kernel estimates, http://www.math.ubc.ca/~barlow/preprints/, 2005.Google Scholar
  6. [6]
    Jean-Michel Bismut, Mécanique aléatoire, Lecture Notes in Mathematics, vol. 866, Springer-Verlag, Berlin, 1981, with an English summary.MATHGoogle Scholar
  7. [7]
    B. Boufoussi and J. van Casteren, An approximation result for a nonlinear Neumann boundary value problem via BSDEs, Stochastic Process. Appl. 114 (2004), no. 2, 331–350.MATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    Brahim Boufoussi, Jan Van Casteren, and N. Mrhardy, Generalized backward doubly stochastic differential equations and SPDEs with nonlinear Neumann boundary conditions, Bernoulli 13 (2007), no. 2, 423–446.MATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    K. Burrage and J.C. Butcher, Stability criteria for implicit Runge-Kutta methods, SIAM J. Numer. Anal. 16 (1979), no. 1, 46–57.MATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    M. Crouzeix, W.H. Hundsdorfer, and M.N. Spijker, On the existence of solutions to the algebraic equations in implicit Runge-Kutta methods, BIT 23 (1983), no. 1, 84–91.MATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    Michel Crouzeix, Sur la B-stabilité des méthodes de Runge-Kutta, Numer. Math. 32 (1979), no. 1, 75–82.MATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    A. Gulisashvili and J.A. van Casteren, Non-autonomous Kato classes and Feynman-Kac propagators, World Scientific, Singapore, 2006.MATHGoogle Scholar
  13. [13]
    E. Hairer and G. Wanner, Solving ordinary differential equations. II, Springer Series in Computational Mathematics, vol. 14, Springer-Verlag, Berlin, 1991, Stiff and differential-algebraic problems.Google Scholar
  14. [14]
    N. Ikeda and S. Watanabe, Stochastic differential equations and diffusion processes, 2 ed., North-Holland Mathematical Library, vol. 24, North-Holland, Amsterdam, 1998.Google Scholar
  15. [15]
    N. El Karoui, É. Pardoux, and M.C. Quenez, Reflected backward SDEs and American options, Numerical methods in finance (L.C.G. Rogers and D. Talay, eds.), Publ. Newton Inst., Cambridge Univ. Press, Cambridge, 1997, pp. 215–231.Google Scholar
  16. [16]
    N. El Karoui and M.C. Quenez, Imperfect markets and backward stochastic differential equations, Numerical methods in finance, Publ. Newton Inst., Cambridge Univ. Press, Cambridge, 1997, pp. 181–214.Google Scholar
  17. [17]
    D. Nualart, The Malliavin calculus and related topics, Probability and its Applications (New York), Springer-Verlag, New York, 1995.Google Scholar
  18. [18]
    _____, Analysis on Wiener space and anticipating stochastic calculus, Lectures on probability theory and statistics (Saint-Flour, 1995), Lecture Notes in Math., vol. 1690, Springer, Berlin, 1998, pp. 123–227.CrossRefGoogle Scholar
  19. [19]
    É. Pardoux, Backward stochastic differential equations and viscosity solutions of systems of semilinear parabolic and elliptic PDEs of second order, Stochastic analysis and related topics, VI (Geilo, 1996), Progr. Probab., vol. 42, Birkhäuser Boston, Boston, MA, 1998, pp. 79–127. MR 99m:35279Google Scholar
  20. [20]
    É. Pardoux and S.G. Peng, Adapted solution of a backward stochastic differential equation, Systems Control Lett. 14 (1990), no. 1, 55–61.MATHCrossRefMathSciNetGoogle Scholar
  21. [21]
    É. Pardoux and S. Zhang, Generalized BSDEs and nonlinear Neumann boundary value problems, Probab. Theory Related Fields 110 (1998), no. 4, 535–558.MATHCrossRefMathSciNetGoogle Scholar
  22. [22]
    Jan Prüss, Maximal regularity for evolution equations in L p -spaces, Conf. Semin. Mat. Univ. Bari (2002), no. 285, 1–39 (2003).Google Scholar
  23. [23]
    D. Revuz and M. Yor, Continuous martingales and Brownian motion, third ed., Springer-Verlag, Berlin, 1999.MATHGoogle Scholar
  24. [24]
    Steven E. Shreve, Stochastic calculus for finance. II, Springer Finance, Springer-Verlag, New York, 2004, Continuous-time models.MATHGoogle Scholar
  25. [25]
    Jan A. Van Casteren, The Hamilton-Jacobi-Bellman equation and the stochastic Noether theorem, Proceedings Conference Evolution Equations 2000: Applications to Physics, Industry, Life Sciences and Economics-EVEQ2000 (M. Iannelli and G. Lumer, eds.), Birkhäuser, 2003, pp. 381–408.Google Scholar
  26. [26]
    _____, Backward stochastic differential equations and Markov processes, Liber Amicorum, Richard Delanghe: een veelzijdig wiskundige (Gent) (F Brackx and H. De Schepper, eds.), Academia Press, University of Gent, 2005, pp. 199–239.Google Scholar
  27. [27]
    _____, Viscosity solutions, backward stochastic differential equations and Markov processes, Preprint University of Antwerp, Technical Report 2006-15, submitted to “Integration: Mathematical Theory and Applications”, 2007.Google Scholar
  28. [28]
    J.-C. Zambrini, A special time-dependent quantum invariant and a general theorem on quantum symmetries, Proceedings of the second International Workshop Stochastic Analysis and Mathematical Physics: ANESTOC’96 (Singapore) (R. Rebolledo, ed.), World Scientific, 1998, Workshop: Vina del Mar Chile 16–20 December 1996, pp. 197–210.Google Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2007

Authors and Affiliations

  • Jan A. Van Casteren
    • 1
  1. 1.Department of Mathematics and Computer ScienceUniversity of AntwerpAntwerpBelgium

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