  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.
