A Generalized It$\hat {\rm o}$'s Formula in Two-Dimensions and   Stochastic Lebesgue-Stieltjes Integrals
  A generalized It${\hat {\rm o}}$ formula for time dependent functions of two-dimensional continuous semi-martingales is proved. The formula uses the local time of each coordinate process of the semi-martingale, left space and time first derivatives and second derivative $\nabla_1^- \nabla_2^-f$ only which are assumed to be of locally bounded variation in certain variables, and stochastic Lebesgue-Stieltjes integrals of two parameters.The two-parameter integral is defined as a natural generalization of the It${\hat {\rm o}}$ integral and Lebesgue-Stieltjes integral through a type of It${\hat {\rm o}}$ isometry formula. 
Chunrong Feng, Huaizhong Zhao