A generalized Itô formula in two-dimensions and stochastic Lebesgue-Stieltjes integrals

Summary. A generalized Ito ̂ formula for time dependent functions of two-dimensional continuous semi-martingales is proved. The formula uses the local time of each co-ordinate process of the semi-martingale, left space and time first derivatives and second derivative ∇−1 ∇−2 f only which are assumed to be of locally bounded varia-tion in certain variables, and stochastic Lebesgue-Stieltjes integrals of two param-eters. The two-parameter integral is defined as a natural generalization of the Itô integral and Lebesgue-Stieltjes integral through a type of Ito ̂ isometry formula