Weak solution of SDE Show that the SDE

$dX_t=dB_t-\frac{Bt}{1-t}dt$ 

has a weak solution on $[0,t]$ for $0\ge t&lt;1$.

I tried it with Girsanov's theorem. But I can't prove that
$E[\exp(\int_0^t\left(\frac{Bs}{1-s}\right)^2ds]&lt;\infty$ which would imply that $Z_t:=\exp(M_t-\frac{1}{2}\langle M\rangle_t)$ is a continuous martingale $\iff E[Z_t]=1 \quad \forall t\ge0.$ So I can't apply Girsanov.
Hope that it help you :to prove this $$E[\exp(\int_0^t\left(\frac{Bs}{1-s}\right)^2ds]&lt;\infty$$ 
note that
    $0&lt;s&lt;t&lt;1 \\$ so  $ \exists n_0: s^{n_0}&lt;\epsilon$
$$\dfrac{B_s}{1-s}=B_s(1+s+s^2+s^3+...)\\\to \\B_s(1+s+s^2+s^3+...)&lt;B_s .M$$
Hope that it help you :to prove this $$E[\exp(\int_0^t\left(\frac{Bs}{1-s}\right)^2ds]&lt;\infty$$ 
note that
    $0&lt;s&lt;t&lt;1 \\$ so  $ \exists n_0: s^{n_0}&lt;\epsilon$
$$\dfrac{B_s}{1-s}=B_s(1+s+s^2+s^3+...)\\\to \\B_s(1+s+s^2+s^3+...)&lt;B_s .M$$