Introduction to the Theory of the It$\hat{o}$-type Stochastic Integrals and Stochastic Differential Equations

Svetlana Janković

The theory of stochastic differential equations, as a part of the general theory of stochastic processe:;, began to develop in the fifties in the discussions of U. Gikhman, and independently of him, K. Ita. The accepted terminology, however, derived from Ita. In his papers [15], [16], [17], for special classes of stochastic processes he introduced the notion of the stochastic integral and of the stochastic differential equation with respect to a Wiener process. Following the classical theory of ordinary differential equations, he proved the fundamental theorem of the existence and uniqueness of solutions and also the Markov property of solutions. From then on this theory has developed in several aspects, mostly induced by mathematical abstractions or by many applications in technical practice, having in mind that a Gaussian white noise could be mathematically interpreted by a Wiener process.