On the existence and unicity of solutions of stochastic integral equations

  • Catherine Doléans-Dade


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
$$X_t = H_t + \int\limits_0^t {f(s,X_{s - } )dM_s + } \int\limits_0^t {g(s,X_{s - } )dA_s }$$
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.


Integral Equation Stochastic Process Probability Theory Mathematical Biology Continuity Condition 
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Copyright information

© Springer-Verlag 1976

Authors and Affiliations

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

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