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
has one and only one solution, provided the random functions f and g satisfy the properties (L) given below, i.e. a Lipschitz condition
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.
Article PDF
Similar content being viewed by others
References
Dellacherie, C.: Capacités et Processus Stochastiques. Berlin-Heidelberg-New York: Springer 1972
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 1970
Kazamaki, N.: On a stochastic Integral Equation with respect to a Weak martingale. TÔhoku Math. J. 26, 53–63 (1974)
Gihman, I. I., Skorohod, A. V.: Stochastic Differential Equations. Berlin-Heidelberg-New York: Springer 1972
McKean, Jr., H. P.: Stochastic Integrals. New York: Academic Press 1969
Meyer, P. A.: Probabilités et Potentiel. Paris: Hermann 1966
Protter, P. E.: On the Existence, Uniqueness, Convergence, and Explosions of solutions of Systems of Stochastic Integral Equations [to appear]
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Doléans-Dade, C. On the existence and unicity of solutions of stochastic integral equations. Z. Wahrscheinlichkeitstheorie verw Gebiete 36, 93–101 (1976). https://doi.org/10.1007/BF00533992
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00533992