SUCCESSIVE-APPROXIMATIONS TO SOLUTIONS OF STOCHASTIC DIFFERENTIAL-EQUATIONS

In the present paper we shall investigate under what conditions the sequence of stochastic processes constructed by the successive approximations converges uniformly to solutions of a stochastic differential equation of Ito type and shall present the local or global existence and uniqueness theorem for solutions of the above mentioned equation under more general conditions. We note that Lemma 3 in this paper is a generalization of Gard's lemma and guarantees the existence of functions which satisfy the conditions of Theorems 2 and 3 in this paper, respectively. Theorem 3 includes as a special case a generalization of Yamada's theorem which is proved by the method of the successive approximations. © 1992.