A Generalized It$\hat{\mathrm{o}}$'s Formula in Two-Dimensions and Stochastic Lebesgue-Stieltjes Integrals

Abstract

A generalized It$\hat{\mathrm{o}}$ 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$\hat{\mathrm{o}}$ integral and Lebesgue-Stieltjes integral through a type of It$\hat{\mathrm{o}}$ isometry formula.

Modify pages 4-9 and add more references