On some properties of sticky Brownian motion

In this paper, we investigate a generalization of Brownian motion, called sticky skew Brownian motion, which has two interesting characteristics: stickiness and skewness. This kind of processes spends a lot more time at its sticky points so that the time they spend at the sticky points has positive Lebesgue measure. By using time change, we obtain an SDE for the sticky skew Brownian motion. Then, we present the explicit relationship between symmetric local time and occupation time. Some basic probability properties, such as transition density, are studied and we derive the explicit expression of Laplace transform of transition density for the sticky skew Brownian motion. We also consider the first hitting time problems over a constant boundary and a random jump boundary, respectively, and give some corollaries based on the results above.