Change of Variables with Local Time on Surfaces for Jump Processes

The `local time on curves' formula of Peskir provides a stochastic change of variables formula for a function whose derivatives may be discontinuous over a time-dependent curve, a setting which occurs often in applications in optimal control and beyond. This formula was further extended to higher dimensions and to include processes with jumps under conditions which may be hard to verify in practice. We build upon the work of Du Toit in weakening the required conditions by allowing semimartingales with jumps. In addition, under vanishing of the sectional first derivative (the so-called `smooth fit' condition), we show that the classical It\^o formula still holds under general conditions.