The Girsanov Theorem Without (So Much) Stochastic Analysis

  • Antoine Lejay
Part of the Lecture Notes in Mathematics book series (LNM, volume 2215)


In this pedagogical note, we construct the semi-group associated to a stochastic differential equation with a constant diffusion and a Lipschitz drift by composing over small times the semi-groups generated respectively by the Brownian motion and the drift part. Similarly to the interpretation of the Feynman-Kac formula through the Trotter-Kato-Lie formula in which the exponential term appears naturally, we construct by doing so an approximation of the exponential weight of the Girsanov theorem. As this approach only relies on the basic properties of the Gaussian distribution, it provides an alternative explanation of the form of the Girsanov weights without referring to a change of measure nor on stochastic calculus.


Girsanov theorem Lie-Trotter-Kato formula Feynman-Kac formula Stochastic differential equation Euler scheme Splitting scheme Flow Heat equation Cameron-Martin theorem 



I wish to thank K. Coulibaly-Pasquier, V. Bally and A. Kohatsu-Higa for some motivating and interesting discussions on this approach. This article is a follow-up of a talk given at the “groupe de travail” of the Probability and Statistics teams of Institut Élie Cartan de Lorraine (Nancy) on the link between the Trotter-Kato-Lie and the Feynman-Kac formula, and I am grateful to the audience for his/her patience.


  1. 1.
    J. Akahori, T. Amaba, S. Uraguchi, An algebraic approach to the Cameron-Martin-Maruyama-Girsanov formula. Math. J. Okayama Univ. 55, 167–190 (2013)MathSciNetzbMATHGoogle Scholar
  2. 2.
    L. Ambrosio, D. Trevisan, Lecture notes on the DiPerna-Lions theory in abstract measure spaces (2015). arXiv: 1505.05292Google Scholar
  3. 3.
    R. Bafico, P. Baldi, Small random perturbations of Peano phenomena. Stochastics 6(3–4), 279–292 (1981/1982). MathSciNetCrossRefGoogle Scholar
  4. 4.
    V. Bally, A. Kohatsu-Higa, A probabilistic interpretation of the parametrix method. Ann. Appl. Probab. 25(6), 3095–3138 (2015). MathSciNetCrossRefGoogle Scholar
  5. 5.
    V. Bally, C. Rey, Approximation of Markov semigroups in total variation distance. Electron. J. Probab. 21, 44 (2016). Paper No. 12.
  6. 6.
    D.R. Bell, The Malliavin Calculus. Pitman Monographs and Surveys in Pure and Applied Mathematics (Longman Scientific & Technical/Wiley, Harlow/New York, 1987)Google Scholar
  7. 7.
    A. Beskos, G.O. Roberts, Exact simulation of diffusions. Ann. Appl. Probab. 15(4), 2422–2444 (2005). MathSciNetCrossRefGoogle Scholar
  8. 8.
    S. Blanes, F. Casas, A. Murua, Splitting and composition methods in the numerical integration of differential equations. Bol. Soc. Esp. Mat. Apl. Se MA 45, 89–145 (2008)Google Scholar
  9. 9.
    R.H. Cameron, W.T. Martin, Transformations of Wiener integrals under translations. Ann. Math. (2) 45, 386–396 (1944)Google Scholar
  10. 10.
    R.H. Cameron, W.T. Martin, Transformations of Wiener integrals under a general class of linear transformations. Trans. Am. Math. Soc. 58, 184–219 (1945)MathSciNetCrossRefGoogle Scholar
  11. 11.
    R.H. Cameron, W.T. Martin, The transformation of Wiener integrals by nonlinear transformations. Trans. Am. Math. Soc. 66, 253–283 (1949). MathSciNetCrossRefGoogle Scholar
  12. 12.
    P.R. Chernoff, Product Formulas, Nonlinear Semigroups, and Addition of Unbounded Operators. Memoirs of the American Mathematical Society, vol. 140 (American Mathematical Society, Providence, 1974)Google Scholar
  13. 13.
    R.J. DiPerna, P.L. Lions, Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98(3), 511–547 (1989)MathSciNetCrossRefGoogle Scholar
  14. 14.
    C. Doléans-Dade, Quelques applications de la formule de changement de variables pour les semimartingales. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 16, 181–194 (1970)MathSciNetCrossRefGoogle Scholar
  15. 15.
    J.L. Doob, Conditional Brownian motion and the boundary limits of harmonic functions. Bull. Soc. Math. France 85, 431–458 (1957)MathSciNetCrossRefGoogle Scholar
  16. 16.
    D. Duffie, P. Protter, From discrete- to continuous-time finance: weak convergence of the financial gain process. J. Math. Finance 2, 1–15 (1992)CrossRefGoogle Scholar
  17. 17.
    K.-J. Engel, R. Nagel, One-Parameter Semigroups for Linear Evolution Equations. Graduate Texts in Mathematics, vol. 194. With contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli, R. Schnaubelt (Springer, New York, 2000)Google Scholar
  18. 18.
    M. Erraoui, E.H. Essaky, Canonical representation for Gaussian processes, in Séminaire de Probabilités XLII. Lecture Notes in Mathematics, vol. 1979 (Springer, Berlin, 2009), pp. 365–381.
  19. 19.
    F. Flandoli, Remarks on uniqueness and strong solutions to deterministic and stochastic differential equations. Metrika 69(2–3), 101–123 (2009). MathSciNetCrossRefGoogle Scholar
  20. 20.
    I.V. Girsanov, On transforming a class of stochastic processes by absolutely continuous substitution of measures. Theor. Probab. Appl. 5, 285–301 (1960). MathSciNetCrossRefGoogle Scholar
  21. 21.
    J.A. Goldstein, Remarks on the Feynman-Kac formula, in Partial Differential Equations and Dynamical Systems. Research Notes in Mathematics, vol. 101 (Pitman, Boston, 1984), pp. 164–180Google Scholar
  22. 22.
    H. Gzyl, The Feynman-Kac formula and the Hamilton-Jacobi equation. J. Math. Anal. Appl. 142(1), 74–82 (1989). MathSciNetCrossRefGoogle Scholar
  23. 23.
    E. Hairer, C. Lubich, G. Wanner, Structure-preserving algorithms for ordinary differential equations, in Geometric Numerical Integration. Springer Series in Computational Mathematics, 2nd edn., vol. 31 (Springer, Berlin, 2006)Google Scholar
  24. 24.
    H. Hult, Approximating some Volterra type stochastic integrals with applications to parameter estimation. Stoch. Process. Appl. 105(1), 1–32 (2003). MathSciNetCrossRefGoogle Scholar
  25. 25.
    M. Jeanblanc, J. Pitman, M. Yor, The Feynman-Kac formula and decomposition of Brownian paths. Mat. Appl. Comput. 16(1), 27–52 (1997)MathSciNetzbMATHGoogle Scholar
  26. 26.
    M. Jeanblanc, M. Yor, M. Chesney, Mathematical Methods for Financial Markets. Springer Finance (Springer, London, 2009). CrossRefGoogle Scholar
  27. 27.
    M. Kac, On distributions of certain Wiener functionals. Trans. Am. Math. Soc. 65, 1–13 (1949)MathSciNetCrossRefGoogle Scholar
  28. 28.
    T. Kato, Trotter’s product formula for an arbitrary pair of self-adjoint contraction semigroups, in Topics in Functional Analysis (Essays Dedicated to M. G. Kreı̆n on the Occasion of his 70th Birthday). Advances in Mathematics Supplied Studies, vol. 3 (Academic, New York, 1978), pp. 185–195Google Scholar
  29. 29.
    A. Kohatsu-Higa, A. Lejay, K. Yasuda, Weak rate of convergence of the Euler-Maruyama scheme for stochastic differential equations with non-regular drift. J. Comput. Appl. Math. 326C, 138–158 (2017). MathSciNetCrossRefGoogle Scholar
  30. 30.
    T.G. Kurtz, P.E. Protter, Weak convergence of stochastic integrals and differential equations, in Probabilistic Models for Nonlinear Partial Differential Equations (Montecatini Terme, 1995). Lecture Notes in Mathematics, vol. 1627 (Springer, Berlin, 1996), pp. 1–41.
  31. 31.
    R. Leandre, Applications of the Malliavin calculus of Bismut type without probability. WSEAS Trans. Math. 5(11), 1205–1210 (2006)MathSciNetzbMATHGoogle Scholar
  32. 32.
    S. Lie, F. Engel, Theorie der Transformationsgruppen (Teubner, Leipzig, 1888)Google Scholar
  33. 33.
    R.S. Liptser, A.N. Shiryaev, Statistics of Random Processes. I. General Theory. Applications of Mathematics (New York), 2nd edn., vol. 5 (Springer, Berlin, 2001). Translated from the 1974 Russian original by A.B. Aries.
  34. 34.
    R.S. Liptser, A.N. Shiryaev, Statistics of Random Processes. II. Applications. Applications of Mathematics (New York), 2nd edn., vol. 6 (Springer, Berlin, 2001). Translated from the 1974 Russian original by A. B. Aries.
  35. 35.
    B. Maisonneuve, Quelques martingales remarquables associées à une martingale continue. Publ. Inst. Statist. Univ. Paris 17.fasc. 3, 13–27 (1968)Google Scholar
  36. 36.
    P. Malliavin, Stochastic Analysis. Grundlehren der Mathematischen Wissenschaften, vol. 313 (Springer, Berlin, 1997).
  37. 37.
    G. Maruyama, On the transition probability functions of the Markov process. Nat. Sci. Rep. Ochanomizu Univ. 5, 10–20 (1954)MathSciNetzbMATHGoogle Scholar
  38. 38.
    R. Mikulevičius, E. Platen, Rate of convergence of the Euler approximation for diffusion processes. Math. Nachr. 151, 233–239 (1991). MathSciNetCrossRefGoogle Scholar
  39. 39.
    R. Mikulevičius, C. Zhang, Weak Euler approximation for Itô diffusion and jump processes. Stoch. Anal. Appl. 33(3), 549–571 (2015). MathSciNetCrossRefGoogle Scholar
  40. 40.
    E. Nelson, Feynman integrals and the Schrödinger equation. J. Math. Phys. 5, 332–343 (1964)CrossRefGoogle Scholar
  41. 41.
    S. Ninomiya, N. Victoir, Weak approximation of stochastic differential equations and application to derivative pricing. Appl. Math. Finance 15(1–2), 107–121 (2008). MathSciNetCrossRefGoogle Scholar
  42. 42.
    S. Orey, Conditions for the absolute continuity of two diffusions. Trans. Am. Math. Soc. 193, 413–426 (1974)MathSciNetCrossRefGoogle Scholar
  43. 43.
    A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences, vol. 44 (Springer, New York, 1983).
  44. 44.
    B. Roynette, M. Yor, Penalising Brownian Paths. Lecture Notes in Mathematics, vol. 1969 (Springer, Berlin, 2009).
  45. 45.
    B. Simon, Schrödinger semigroups. Bull. Am. Math. Soc. (N.S.) 7(3), 447–526 (1982).
  46. 46.
    A.V. Skorohod, On the densities of probability measures in functional spaces, in Proceedings of. Fifth Berkeley Symposium on Mathematical Statistics and Probability (Berkeley, CA, 1965/1966) (University of California Press, Berkeley, 1967); vol. II: Contributions to Probability Theory, Part 1Google Scholar
  47. 47.
    D.W. Stroock, S.R.S. Varadhan, Multidimensional Diffusion Processes. Classics in Mathematics (Springer, Berlin, 2006). Reprint of the 1997 edn.Google Scholar
  48. 48.
    S. Takanobu, On the error estimate of the integral kernel for the Trotter product formula for Schrödinger operators. Ann. Probab. 25(4), 1895–1952 (1997). MathSciNetCrossRefGoogle Scholar
  49. 49.
    D. Talay, Discrétisation d’une équation différentielle stochastique et calcul approché d’espérances de fonctionnelles de la solution. RAIRO Modél. Math. Anal. Numér. 20(1), 141–179 (1986)MathSciNetCrossRefGoogle Scholar
  50. 50.
    H.F. Trotter, On the product of semi-groups of operators. Proc. Am. Math. Soc. 10, 545–551 (1959)MathSciNetCrossRefGoogle Scholar
  51. 51.
    J. H. Van Schuppen, E. Wong, Transformation of local martingales under a change of law. Ann. Probab. 2, 879–888 (1974)MathSciNetCrossRefGoogle Scholar
  52. 52.
    A.J. Veretennikov, On strong solutions and explicit formulas for solutions of stochastic integral equations. Math. USSR Sb. 39(3), 387–403 (1981). CrossRefGoogle Scholar
  53. 53.
    A. Zvonkin, A transformation of the phase space of a diffusion process that removes the drift. Math. USSR Sb. 22, 129–149 (1975). CrossRefGoogle Scholar

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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Université de Lorraine, CNRS, Inria, IECLNancyFrance

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