I was wondering how to adjust the self financing condition when cash borrowing cas be secured by the stock.
Suppose the risk-free money account is $B_t$ and there is a risky asset $S_t$. One have that $dB_t=r_tB_tdt$ where $r_t$ is the risk free instantaneous rate.
In the classical seeting, the strategy $\pi_t = b_tB_t +\varphi_tS_t$ is self financed if $d\pi_t = b_t dB_t + \varphi_t dS_t=b_tr_tB_tdt + \varphi_t dS_t$.
Suppose now that I start from $t=0$ with an empty portfolio, I then borrow $S_t$, buy a stock and pledge stock to secure the loan. At $t+dt$, I would have to pay $(r_t - q_t)S_t dt$ worth of interests on the loan, where $q_t$ is the stock's repo rate.
What would be $(b_t,\varphi_t)$ in that case and where should I account for the repo rate (my guess is that it is in the stock's drift : $dS_t=(r_t - q_t)dt + \sigma_t dW^Q_t$) ? Should one add a new money account for stock secured funding ?