A CLASS OF STOCHASTIC DIFFERENTIAL EQUATIONS WITH LOCAL TIME AND SKEW BROWNIAN MOTION WITH TWO BOUNDARIES

Consider the following stochastic differential equation: d X(t)=\sigma\left(X_t\right) d B(t)+b\left(X_t\right) d t+\beta_1 d \hatL_t^0(X)+\beta_2 d \hatL_t^1(X) where B(t) is a normalized Brownian motion, \beta_1 and \beta_2 are constants, \hatL_t^*(\boldsymbolX) is the symmetric local time at x of X, the following theorem is established: Theorem 1: If the following conditions are satisfied: (1) \left|\beta_1\right|< 1,\left|\beta_2\right|< 1; (2) \sigma: R \rightarrow R, \sigma(x) is Borel measurable, \forall x \in R_7 \sigma(x) \neq 0, and for any u, v_;-\infty< u< v< \infty, we hewe: \int_u^0 \frac1\sigma^2(x) d x< \infty, \int_u^v \frac|b(x)|\sigma^2(x) d x< \infty; (3) \int_-\infty^0 \exp \left\\int_y^0 \frac2 b(z)\sigma^2(z) d z\right\ d y=\int_0^\infty \exp \left\-\int_0^\nu \frac2 b(z)\sigma^2(z) d z\right\ d y=\infty, then there exists a unique solution X of equation (1) X is called a skew Brownian motion with two boundaries, if d X(t)=d B(t)+\beta_1 d \hatL_i^0(X)+\beta_2 d \hatL_t^1(X). In this paper, the construction and martingale charaoterization of skew Brownian motions with two boundaries are established, too.