Probability Theory and Related Fields

, Volume 151, Issue 3–4, pp 613–657 | Cite as

Integration by parts formula and applications to equations with jumps

  • Vlad BallyEmail author
  • Emmanuelle Clément


We establish an integration by parts formula in an abstract framework in order to study the regularity of the law for processes arising as the solution of stochastic differential equations with jumps, including equations with discontinuous coefficients for which the Malliavin calculus developed by Bichteler et al. (Stochastics Monographs, vol 2. Gordon & Breach, New York, 1987) and Bismut (Z Wahrsch Verw Gebiete 63(2):147–235, 1983) fails.


Integration by parts formula Malliavin calculus Stochastic equations Poisson point measures 

Mathematics Subject Classification (2000)

Primary 60H07 Secondary 60G51 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Applebaum D.: Universal Malliavin calculus in Fock and Lévy-Itô spaces. Commun. Stoch. Anal. 3(1), 119–141 (2009)MathSciNetGoogle Scholar
  2. 2.
    Bally V., Bavouzet M.-P., Messaoud M.: Integration by parts formula for locally smooth laws and applications to sensitivity computations. Ann. Appl. Probab. 17(1), 33–66 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Bally, V., Fournier, N.: Regularization properties of the 2d homogeneous Boltzmann equation without cutoff (2009, preprint)Google Scholar
  4. 4.
    Bass R.F.: Uniqueness in law for pure jump Markov processes. Probab. Theory Relat. Fields 79(2), 271–287 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Bavouzet-Morel, M.-P., Messaoud, M.: Computation of Greeks using Malliavin’s calculus in jump type market models. Electron. J. Probab. 11(10), 276–300 (electronic) (2006)Google Scholar
  6. 6.
    Bichteler, K., Gravereaux, J.-B., Jacod, J.: Malliavin calculus for processes with jumps. In: Stochastics Monographs, vol. 2. Gordon & Breach, New York (1987)Google Scholar
  7. 7.
    Bismut J.-M.: Calcul des variations stochastique et processus de sauts. Z. Wahrsch. Verw. Gebiete 63(2), 147–235 (1983)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Nunno, G.D., Øksendal, B., Proske, F.: Malliavin calculus for Lévy processes with applications to finance. In: Universitext. Springer, Berlin (2009)Google Scholar
  9. 9.
    Fournier N.: Jumping SDEs: absolute continuity using monotonicity. Stoch. Process. Appl. 98(2), 317–330 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Fournier N.: Smoothness of the law of some one-dimensional jumping S.D.E.s with non-constant rate of jump. Electron. J. Probab. 13(6), 135–156 (2008)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Fournier N., Giet J.-S.: Existence of densities for jumping stochastic differential equations. Stoch. Process. Appl. 116(4), 643–661 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Ikeda, N., Watanabe, S.: Stochastic differential equations and diffusion processes. In: North-Holland Mathematical Library, 2nd edn, vol. 24. North-Holland, Amsterdam (1989)Google Scholar
  13. 13.
    Ishikawa Y., Kunita H.: Malliavin calculus on the Wiener–Poisson space and its application to canonical SDE with jumps. Stoch. Process. Appl. 116(12), 1743–1769 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Jacob, N.: Pseudo differential operators and Markov processes. vol. I. In: Fourier Analysis and Semigroups. Imperial College Press, London (2001)Google Scholar
  15. 15.
    Jacob, N.: Pseudo differential operators and Markov processes. vol. II. In: Generators and their Potential Theory. Imperial College Press, London (2002)Google Scholar
  16. 16.
    Jacob, N.: Pseudo differential operators and Markov processes. vol. III. In: Markov Processes and Applications. Imperial College Press, London (2005)Google Scholar
  17. 17.
    Kolokoltsov V.N.: On Markov processes with decomposable pseudo-differential generators. Stoch. Stoch. Rep. 76(1), 1–44 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Kolokoltsov, V.N.: The Lévy–Khintchine type operators with variable lipschitz continuous coefficients generate linear or nonlinear markov processes and semigroups. PTRF (1) (2009, to appear)Google Scholar
  19. 19.
    Nourdin I., Simon T.: On the absolute continuity of Lévy processes with drift. Ann. Probab. 34(3), 1035–1051 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Nualart, D.: Analysis on Wiener space and anticipating stochastic calculus. In: Lectures on Probability Theory and Statistics (Saint-Flour, 1995). Lecture Notes in Mathematics, vol. 1690, pp. 123–227. Springer, Berlin (1998)Google Scholar
  21. 21.
    Picard J.: On the existence of smooth densities for jump processes. Probab. Theory Relat. Fields 105(4), 481–511 (1996)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Laboratoire d’Analyse et de Mathématiques Appliquées, Université Paris-Est, UMR 8050Marne-la-Vallée Cedex 2France

Personalised recommendations