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.
Categories: math.PR