 Solution

Write $T=t_0$ and $B_t=e^{-\rho (T-t)}$ for the risk-free asset paying one at $T$. Under the physical measure, $d{S}_t=\mu S_{t}dt+\sigma S_{t}d{W}_t$, with $\sigma >0$, while $d{B}_t=\rho B_{t}dt$. A self-financing portfolio with predictable stock and bond holdings $g_t,h_t$ has value $V_t=g_t{S}_t+h_t{B}_t$ and gainsRebalancing therefore adds no outside money and withdraws none. This gains equation, rather than merely the value decomposition, defines self-financing.

Let $p(x,t)=xg(x,t)+B_{t}h(x,t)$. The Itô formula givesThe prescribed portfolio gains instead have diffusion coefficient $\sigma S_{t}g$ and drift $\mu S_{t}g+\rho B_{t}h$. Equality of diffusion coefficients requires $p_x=g$, equivalentlyUnder this condition, $p_{xx}=g_x$. Equality of drifts is then $p_t+\frac{1}{2}{\sigma }^2{x}^2{g}_x=\rho B_{t}h$. Since $p_t=x{g}_t+B_t{h}_t+\rho B_{t}h$, this becomesThese conditions are also sufficient by the same Itô formula. Necessity initially holds along the stock process. Its positive transition density on $(0,\infty )$ for every positive time, together with continuity of the coefficient functions, extends the identities to all $x>0$ and then to the time endpoints by continuity.

For a self-financing portfolio, substitute $g=p_x$ and $B_{t}h=p-x{p}_x$ into the drift identity to obtain the Black-Scholes equationConversely, a smooth solution of this equation gives a self-financing portfolio by choosing the delta hedgeIts value is $p$, and both coefficient identities follow directly.

The printed deduction needs this holdings qualification: the Black-Scholes value equation needs delta-compatible holdings. A value function alone cannot make an arbitrary prescribed decomposition self-financing. As a counterexample, take $p=0$, $g=1$, $h=-x/B_t$. The value function solves the Black-Scholes equation, but this zero-value portfolio has gains $d{S}_t-\rho S_{t}dt$, which contain the nonzero term $\sigma S_{t}d{W}_t$. Thus for the originally specified $g,h$, the exact equivalence is the pricing equation together with $g=p_x$, not the pricing equation alone.

For a claim paying the holder at rate $k(S_t)$, let $p$ denote its remaining, ex-dividend value. A replicating portfolio pays out $k(S_t)dt$, so its net wealth equation is $dp=gdS+hdB-k(S_t)dt$. The delta hedge is still $g=p_x$, but the Black-Scholes equation becomesIf there is a terminal payoff $f$, set $p(x,T)=f(x)$; with no terminal payment set it to zero. The corresponding risk-neutral valuation isEach intervening payment is discounted from its own payment time.