Stochastic Differential Equations

  • Daniel Revuz
  • Marc Yor
Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 293)

Abstract

In previous chapters stochastic differential equations have been mentioned several times in an informal manner. For instance, if M is a continuous local martingale, its exponential ε(M) satisfies the equality
$$\mathcal{E}{(M)_t} = 1 + \int_0^t {\mathcal{E}{{(M)}_s}} d{M_s};$$
this can be stated: ε(M) is a solution to the stochastic differential equation
$${X_t} = 1 + \int_0^t {{X_s}d{M_s},} $$
which may be written in differential form
$$d{X_t} = {X_t}d{M_t},{X_0} = 1.$$
We have even seen (Exercise (3.10) Chap. IV) that ε(M) is the only solution to this equation. Likewise we saw in Sect. 2 Chap. VII, that some Markov processes are solutions of what may be termed stochastic differential equations.

Keywords

Brownian Motion Stochastic Differential Equation Strong Solution Predictable Function Local Martingale 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes and Comments

  1. 2.
    Itô, K. tochastic differential equations. Memoirs A.M.S. 4, 1951Google Scholar
  2. 1.
    Ikeda, N., and Watanabe, S. comparison theorem for solutions of stochastic differential equations and its applications. Osaka J. Math. 14 (1977) 619 - 633MathSciNetMATHGoogle Scholar
  3. 5.
    Perkins, E. Local time and pathwise uniqueness for stochastic differential equations. Sém. Prob. XVI. Lecture Notes in Mathematics, vol. 920. Springer, Berlin Heidelberg New York 1982, pp. 201208Google Scholar
  4. 1.
    Kawabata, S., and Yamada, T. On some limit theorems for solutions of stochastic differential equations. Sém. Prob. XVI. Lecture Notes in Mathematics, vol. 920. Springer, Berlin Heidelberg New York 1982, pp. 412 - 441Google Scholar
  5. 1.
    Biane, P., Le Gall, J.F., and Yor, M. Un processus qui ressemble au pont brownien. Sém. Prob. XXI. Lecture Notes in Mathematics, vol. 1247. Springer, Berlin Heidelberg New York 1987, pp. 270 - 275Google Scholar
  6. 1.
    Stroock, D.W., and Yor, M. On extremal solutions of martingale problems. Ann. Sci. Ecole Norm. Sup. 13 (1980) 95 - 164MathSciNetMATHGoogle Scholar
  7. 1.
    Liptser, R.S., and Shiryaev, A.N. Statistics of random processes I and II. Springer Verlag, Berlin, 1977 and 1978Google Scholar
  8. 1.
    Benés, V. Non existence of strong non-anticipating solutions to SDE’s; Implications for functional DE’s, filtering and control. Stoch. Proc. Appl. 5 (1977) 243 - 263CrossRefGoogle Scholar
  9. 1.
    Biane, P., Le Gall, J.F., and Yor, M. Un processus qui ressemble au pont brownien. Sém. Prob. XXI. Lecture Notes in Mathematics, vol. 1247. Springer, Berlin Heidelberg New York 1987, pp. 270 - 275Google Scholar
  10. 9.
    Yor, M. Sur l’étude des martingales continues extrémales. Stochastics 2 (1979) 191 - 196MathSciNetMATHCrossRefGoogle Scholar
  11. 4.
    Yamada, T. On some limit theorems for occupation times of one-dimensional Brownian motion and its continuous additive functionals locally of zero energy. J. Math. Kyoto Univ. 26, 2 (1986) 309 - 222MathSciNetMATHGoogle Scholar
  12. 1.
    Ikeda, N., and Watanabe, S. comparison theorem for solutions of stochastic differential equations and its applications. Osaka J. Math. 14 (1977) 619 - 633MathSciNetMATHGoogle Scholar
  13. 2.
    Ikeda, N., and Watanabe, S. tochastic differential equations and diffusion processes. North-Holland Publ. Co., Amsterdam Oxford New York; Kodansha Ltd., Tokyo 1981Google Scholar
  14. 1.
    Le Gall, J.F. Applications du temps local aux équations différentielles stochastiques unidimensionnelles. Sém. Prob. XVII. Lecture Notes in Mathematics, vol. 986. Springer, Berlin Heidelberg New York 1983, pp. 15 - 31Google Scholar
  15. 4.
    Williams, D. A simple geometric proof of Spitzer’s winding number formula for 2-dimensional Brownian motion. University College, Swansea (1974)Google Scholar
  16. 11.
    Yor, M. Remarques sur une formule de P. Lévy. Sém. Prob. XIV. Lecture Notes in Mathematics, vol. 784. Springer, Berlin Heidelberg New York 1980, pp. 343 - 346Google Scholar
  17. 2.
    Stroock, D.W., and Yor, M. Some remarkable martingales. Sém. Prob. XV. Lecture Notes in Mathematics, vol. 850. Springer, Berlin Heidelberg New York 1981, pp. 590 - 603Google Scholar
  18. 7.
    Yor, M. Les filtrations de certaines martingales du mouvement brownien dans R". Sém. Prob. XIII. Lecture Notes in Mathematics, vol. 721. Springer, Berlin Heidelberg New York 1979, pp. 427440Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1991

Authors and Affiliations

  • Daniel Revuz
    • 1
  • Marc Yor
    • 2
  1. 1.Département de MathématiquesUniversité de Paris VIIParis Cedex 05France
  2. 2.Laboratoire de ProbabilitésUniversité Pierre et Marie CurieParis Cedex 05France

Personalised recommendations