For a market with a bank and risky assets $S_1, S_2$ with different volatility, what should be the short interest rate in this market?

Question

Let there be two assets $S_1$ and $S_2$ s.t.for $\sigma_1 \neq \sigma_2$ $$dS_{1t}=\mu_1 S_{1t}dt+ \sigma_1S_{1t}dB_t \\dS_{2t}=\mu_2 S_{2t}dt+ \sigma_2 S_{2t}dB_t$$ . If there exists a bank, what should be the short interest rate in this market ?

I have tried to make use of the following argument;

If $V_t$ is a riskless self-financing portfolio, then $dV_t=rV_t dt$ must be satisfied where $r$ is the short interest rate.

I want to make a riskless self-financing portfolio out of $S_1,S_2,G_t$ where $G_t=e^{rt}$ so put $V_t=a_t S_{1t}+b_tS_{2t}+c_tG_t$. I tried to calculate the differential but couldn't get any useful result.

Any help is appreciated.

Answer 1

Thanks to @Antoine Conze, here's my answer.

Using self-finance condition, straight forward calculation shows, with some sloppy notation, $$dV=(a\mu_1S_1+b\mu_2 S_2+crG)dt+(a\sigma_1 S_1+b\sigma_2S_2)dB =r(aS_1+bS_2+cG)dt\\ \Rightarrow a(\mu_1-r)S_1+b(\mu_2 -r)S_2=0,\ a\sigma_1S_1 +b \sigma_2 S_2=0 \\ \Rightarrow r=\frac{\mu_1\sigma_2-\mu_2\sigma_1}{\sigma_2 - \sigma_1} \text{ comparing the coefficients }$$.