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On the existence and unicity of solutions of stochastic integral equations

  • Catherine Doléans-Dade
Article

Abstract

Let M be a local martingale, A be an adapted process with finite variation on each finite interval and H be an adapted cadlag process (i.e. H is continuous on the right and has finite left limits). We shall prove that the equation
$$X_t = H_t + \int\limits_0^t {f(s,X_{s - } )dM_s + } \int\limits_0^t {g(s,X_{s - } )dA_s }$$
(1)
has one and only one solution, provided the random functions f and g satisfy the properties (L) given below, i.e. a Lipschitz condition
$$|g(s,\omega ,x) - g(s,\omega ,y)| + |f(s,\omega ,x) - f(s,\omega ,y)|\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } K|x - y|,$$
and two less stringent properties.

Results of this kind were proved recently by Kazamaki (3) and Protter (7) under much more restrictive continuity conditions on M and A.

Keywords

Integral Equation Stochastic Process Probability Theory Mathematical Biology Continuity Condition 
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.

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References

  1. 1.
    Dellacherie, C.: Capacités et Processus Stochastiques. Berlin-Heidelberg-New York: Springer 1972Google Scholar
  2. 2.
    Doléans-Dade, C., Meyer, P. A.: Intégrales Stochastiques par rapport aux martingales locales. Lecture Notes in Math. 124, p. 77–107. Berlin-Heidelberg-New York: Springer 1970Google Scholar
  3. 3.
    Kazamaki, N.: On a stochastic Integral Equation with respect to a Weak martingale. TÔhoku Math. J. 26, 53–63 (1974)Google Scholar
  4. 4.
    Gihman, I. I., Skorohod, A. V.: Stochastic Differential Equations. Berlin-Heidelberg-New York: Springer 1972Google Scholar
  5. 5.
    McKean, Jr., H. P.: Stochastic Integrals. New York: Academic Press 1969Google Scholar
  6. 6.
    Meyer, P. A.: Probabilités et Potentiel. Paris: Hermann 1966Google Scholar
  7. 7.
    Protter, P. E.: On the Existence, Uniqueness, Convergence, and Explosions of solutions of Systems of Stochastic Integral Equations [to appear]Google Scholar

Copyright information

© Springer-Verlag 1976

Authors and Affiliations

  • Catherine Doléans-Dade
    • 1
  1. 1.Department of MathematicsUniversity of Illinois UrbanaUSA

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