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.