EXISTENCE-UNIQUENESS PROBLEMS FOR INFINITE DIMENSIONAL STOCHASTIC DIFFERENTIAL EQUATIONS WITH DELAYS

The main aim of this paper is to develop the basic theory of a class of infinite dimensional stochastic differential equations with delays (IDSDEs) under local Lipschitz conditions. Firstly, we establish a global existence-uniqueness theorem for the IDSDEs under the global Lipschitz condition in \(C\) without the linear growth condition. Secondly,  the non-continuable solution  for IDSDEs is given under the local Lipschitz condition in \(C\). Then, the classical Ito's formula is improved  and a  global existence theorem for IDSDEs is obtained. Our new theorems give better results while conditions imposed are much weaker than some existing results. For example, we need only the local Lipschitz condition in \(C\) but neither the linear growth condition nor the continuous condition on the time \(t\). Finally, two examples are provided to show the effectiveness of the theoretical results.