I would like to find how much error I make when I hedge a call option using Black Scholes model in a market which is actually governed by a stochastic volatility process such as $$dS_t = rS_tdt + \sigma_t S_tdW_t^Q$$ where $(\sigma_t)_{t\geq 0}$ is some stochastic process and $(W_t^Q)_{t\geq 0}$ is Brownian motion under the risk neutral measure. Just for the sake of clarity the Black Scholes model is given by $$dS_t = rS_tdt + \sigma_{BS} S_tdW_t^Q$$ where $\sigma_{BS}$ is some known constant.
First I want to determine how my hedging portfolio evolves. I denote this by $V$. There are three things I need to impose on this portfolio. The number of shares in it is given by the Black Scholes delta (so I calculate it based on $\sigma_{BS}$), the portfolio is self-financing and its value at inception has to be equal to the Black Scholes price of the call option. Then it must hold that $$dV_t = \Delta_{BS} dS_t + (V_t - \Delta_{BS} S_t)rdt \qquad V_0 = C_0$$ This then yields $$dV_t = rV_tdt + \Delta_{BS}\sigma_t S_tdW_t^Q \tag{1}$$
On the other hand the price of the call option under the Black Scholes model obeys the SDE $$dC_t = \underbrace{\left(\theta_{BS} + \Delta_{BS}rS_t + \frac{1}{2}\Gamma_{BS}\sigma_{BS}^2S_t^2\right)dt}_{rC_tdt} + \Delta_{BS}\sigma_{BS}S_tdW_t^Q \tag{2}$$
I define the hedging error as $e_t = V_t - C_t$ and I am interested in $e_T$ where $T$ is the expiry of the call option.
I can subtract $(2)$ from $(1)$ but the terms with Brownian motion do not vanish and according to the notes I am looking at I should get something like $\int_0^T\Gamma_{BS}(\sigma_{BS}^2 - \sigma_t^2)\ldots dt$. I am not sure if I defined the objects the right way. Some help would be appreciated.
More details on my reasoning: Let me phrase my question in a different way. Perhaps I am doing something wrong while converting my interpretation into mathematical statements. Just for the sake of argument forget the fact that the Black Scholes model exists and we are given that the market is governed by an SV model. At time $0$ I sell a call option (strike $K$, maturity $T$) in this market and I want to hedge it. Now instead of hedging correctly, I hold $\Delta^{wrong}$ number of stocks at each time where $$\Delta^{wrong}_t = \frac{\log\left(\frac{S_t}{K}\right) + \left(r + \frac{1}{2}\sigma_{wrong}^2\right)(T-t)}{\sigma_{wrong}\sqrt{T-t}}$$ with $\sigma_{wrong}$ being some constant positive number that I pick.
With the money I get from the sale, $V_0$, I set up my porfolio by buying $\Delta^{wrong}_0$ number of shares and borrowing/lending $V_0 - \Delta^{wrong}_0S_0$ in the money markets. Since I want my portfolio to be self-financing, I rebalance it such that
$$dV_t = \Delta_{wrong} dS_t + (V_t - \Delta_{wrong} S_t)rdt$$
Under the SV model, this becomes $$dV_t = rV_tdt + \Delta_{wrong}\sigma_tS_tdW_t$$
If instead of choosing some constant $\sigma_{wrong}$ in $\Delta^{wrong}_t$, I had made the right hedging decisions, it would follow that at maturity $V_T = \max(S_T-K,0)$. But since I hedged incorrectly, it won't be true that $V_T = \max(S_T-K,0)$ and that difference is the hedging error in my interpretation.