Weak Smoothness Conditions for the Change of Variable Formula with Local Time on Surfaces

We study the change of variable formula from [9, 10] in the setting where X = (X1 t , . . . , X n t )t≥0 is a continuous Rn valued semimartingale of which the first n− 1 components are of bounded variation. Given two (or more) continuous surfaces b1, b2 : Rn−1 → R that could intersect, let F : Rn → R be a continuous function which is C1,...,1,2 at all points x ∈ Rn such that xn 6= b1(x1, . . . , xn−1) or xn 6= b2(x1, . . . , xn−1) . The function F is not assumed to be smoothly extendable onto these surfaces. In this setting, we derive sufficient conditions under which F satisfies the change of variable formula from [9, 10]. The sufficient conditions are geared towards applications, and are well-suited to optimal stopping problems for higher dimensional diffusions, or diffusions whose generators have non-smooth or discontinuous coefficients. The results may also prove useful in other areas of research.