An Estimate of Stochastic Integrals and Its Applications

Let （X_t） be the solution of S.D.E.d X_t=\sigma\left(x_t\right) d W_t+b\left(X_t\right) d t, \quad X_0=x, In this paper we obtained an estimate of the form E^x \int_s^t f\left(X_u\right) d u \leq C\prime(t-s)^1 / 2\|f\|, and a new result on existence of soutions of S. D. E