= Solution
Differentiate the backward equation for $V$ with respect to the <stock> state, and put $w=V_S$. The product derivative of $a^2/2$ is $aa'$, giving
$$
w_t+aa'w_S+\frac12a^2w_{SS}=0,\qquad w(T,S)=g'(S).
$$
The assumed smoothness and bounded derivatives put $w$ in the uniqueness class for the equation defining $U$. Therefore $U=V_S$, and $\pi_t=V_S(t,S_t)$ is the <option delta>. Substituting in the <Itô integral> from part (a) and using $dS_t=a(S_t)dW_t$ gives \b[the self-financing representation]
$$
\boxed{\xi_t=V(0,S_0)+\int_0^t\pi_s\,dS_s,\qquad\pi_t=V_S(t,S_t).}
$$
The boundedness hypotheses justify both <stochastic integrals>. This derivation does not divide by $a$ and remains valid even at states with zero <diffusion amplitude>.
Back to article page
