A generalized Ito's formula in two-dimensions and stochastic Lebesgue-Stieltjes integrals

In this paper, a generalized Ito's formula for continuous functions of two-dimensional continuous semimartingales is proved. The formula uses the local time of each coordinate process of the semimartingale, the left space first derivatives and the second derivative del(-)(1)del(-)(2)f, and the stochastic Lebesgue-Stieltjes integrals of two parameters. The second derivative del(-)(1)del(-)(2)f is only assumed to be of locally bounded variation in certain variables. Integration by parts formulae are asserted for the integrals of local times. The two-parameter integral is defined as a natural generalization of both the Ito integral and the Lebesgue-Stieltjes integral through a type of Ito isometry formula.