Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 211 / 6 / a / Solution

Apply the Itô formula to ${\xi }_t=V(t,S_t)$. The backward equation cancels its drift, leavingBounded $a$ and $V_S$ make this Itô integral square-integrable on the finite horizon; in addition $V$ itself is bounded. Hence $\xi$ is a true martingale, not just a local martingale, and ${\xi }_T=g(S_T)$. Taking conditional expectations givesThe stock is understood to be adapted to the stated Brownian filtration, with its initial value fixed there. This is the Feynman-Kac formula in the zero-potential, zero-drift case.