On the existence and unicity of solutions of stochastic integral equations

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.