Itô's Lemma

$$V_t-V_0 = \int_0^t f_x(S_u, T-u)\,dS_u-\int_0^t f_\tau(S_u, ... ...xtstyle{1\over 2}}\sigma^2\int_0^t {S_u}^2f_{xx}(S_u,T-u)\,du,$$

where $V_t=f(S_t,\tau)$ for $0\leq\tau\equiv T-t\leq T$, and $f\in C^{2,1}((0,\infty)\times [0,T])$.

References

Price, J. F. ``Optional Mathematics is Not Optional.'' Not. Amer. Math. Soc. 43, 964-971, 1996.