$a_t S_t$ = number of shares ($S_t$ is stock price at $t$), $S_0 = 1$
$b_t \beta _t$ = saving account value , $d \beta_t = r \beta_t dt$, $r=$ interest rate
So the value of the portfolio:
$$V_t = a_t S_t + b_t \beta_t$$
Is self-financing if
$$dV_t = a_t dS_t + b_t d \beta_t$$
If $a_t = 1-t$, how can I choose $b_t$ such that my portfolio is self-financing?
$$V_t = (1-t)S_t + b_t \beta_t$$
How do I formulate $dV_t$ now? Don't I require more information, in particular, what is $S_t$?
Is there a need to use the stochastic product rule?