A Change-of-Variable Formula with Local Time on Surfaces

Let X = (X; . . . ; X) be a continuous semimartingale and let b : IR 1 ! IR be a continuous function such that the process b = b(X; . . . ; X ) is a semimartingale. Setting C = f (x1; . . . ; xn) 2 IR j xn < b(x1; . . . ; xn 1) g and D = f (x1; . . . ; xn) 2 IR j xn > b(x1; . . . ; xn 1) g suppose that a continuous function F : IR ! IR is given such that F is C1n on C and F is C1 ;...;in on D where each ik equals 1 or 2 depending on whether X is of bounded variation or not. Then the following change-of-variable formula holds: