 Solution

Invest each cash payment $q_v{S}_{v}dv$ received at time $v$ in the bank account until $T$. Its accumulated terminal amount is $q_v{S}_v{e}^{r(T-v)}dv$, so the entire future stream is equivalent, path by path, to the terminal payment ${\int }_t^T{q}_v{S}_v{e}^{r(T-v)}dv$.

The risk-neutral pricing formula, including this reinvested payoff, iswhere $d{\tilde{S}}_v=r{\tilde{S}}_{v}dv+\sigma {\tilde{S}}_{v}d{W}_v^{\mathbb{Q}}$. Equivalently, discount each cash flow directly to time $t$. This is the discounted-gains martingale principle underlying the PDE in part (a).