= Solution
Apply the <Itô formula> to $\xi_t=V(t,S_t)$. The backward equation cancels its <drift>, leaving
$$
d\xi_t=\left(V_t+\frac12a(S_t)^2V_{SS}\right)(t,S_t)dt
+a(S_t)V_S(t,S_t)dW_t
=a(S_t)V_S(t,S_t)dW_t.
$$
Bounded $a$ 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 $\xi_T=g(S_T)$. \b[Taking conditional expectations gives]
$$
\boxed{\xi_t=\mathbb E[g(S_T)\mid\mathcal F_t].}
$$
The <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.
Back to article page
