I am well aware of how the ordinary Black and Scholes equation is derived, under the assumption of an arbitrage free portfolio, $V=G-hS$. Here $S$ is the price of the underlying and $G$ is the option price. Suppose now that the portfolio is stochastic, and follows the process $$(1): dV=rVdt+f(V,t)VdW,$$ where $r$ is a constant parameter. The change of the portfolio also satisfies $dV=dG-h \cdot dS$. In the standard Black and Scholes, we have $h = \partial G/\partial S$. Assuming that the underlying follows a GBM, we have $$dS = \mu S dt + \sigma SdW,$$ where $\mu$ and $\sigma$ are constant parameters. Itô's lemma then provides $$dG = \left(\frac{\partial G}{\partial t}+\mu S\frac{\partial G}{\partial S}+\frac{1}{2}\sigma^2S^2\frac{\partial G^2}{\partial S^2} \right)dt + \sigma S\frac{\partial G}{\partial S}dW,$$ so that $$(2): dV=dG-hdS = \left(\frac{\partial G}{\partial t}+\mu S\left(\frac{\partial G}{\partial S}-h\right)+\frac{1}{2}\sigma^2S^2\frac{\partial G^2}{\partial S^2} \right)dt + \sigma S\left(\frac{\partial G}{\partial S}-h\right)dW $$
Matching 1) and 2), we see that we must have the following implicit relation for $h$ $$(3): f(G-hS,t)(G-hS)=\sigma S \left(\frac{\partial G}{\partial S}-h\right). $$ In the standard Black and Scholes we would have $f=0$, thus recovering $h = \frac{\partial G}{\partial S}$. If $f$ only depends on its second argument ($t$), then 3) can also easily be solved for $h$, yielding $$(5):h =\frac{Gf(t)-\sigma S \frac{\partial G}{\partial S}}{S(f(t)-\sigma)}, \ f \ \mathrm{independent \ of} \ V$$
Combining 1) and 2) would then give the Black and Scholes like equation $$(6):\frac{\partial G}{\partial t}+\mu S\frac{\partial G}{\partial S}+\frac{1}{2}\sigma^2S^2\frac{\partial G^2}{\partial S^2}+(r-\mu)\frac{Gf(t)-\sigma S \frac{\partial G}{\partial S}}{f(t)-\sigma}-rG=0, \ f \ \mathrm{independent \ of} \ V $$
However, what could we do if $f$ is not independent of $V$?