A class of stochastic differential equations with the time average

Stochastic differential equations with the time average have received increasing attentions in recent years since they can offer better explanations for some financial models. Since the time average is involved in this class of stochastic differential equations, in this paper, the linear growth condition and the Lipschitz condition are different from the classical conditions. Under the special linear growth condition and the special Lipschitz condition, this paper establishes the existence and uniqueness of the solution. By using the Lyapunov function, this paper also establishes the existence and uniqueness under the local Lipschitz condition and gives the p-th moment estimate. Finally, a scalar example is given to illustrate the applications of our results.