A Generalized It o's Formula in Two-Dimensions and Stochastic Lebesgue-Stieltjes Integrals
Chunrong Feng, Huaizhong Zhao
Abstract
A generalized It 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 ∇1- ∇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 o integral and Lebesgue-Stieltjes integral through a type of It o isometry formula.
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