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Show that the SDE

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

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

I tried it with Girsanov's theorem. But I can't prove that $E[\exp(\int_0^t\left(\frac{Bs}{1-s}\right)^2ds]<\infty$ 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.

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    I guess there is a typo in your question, because that's actually not really an SDE ... there is no $X_t$ on the right-hand side of your equation. – 2017-02-17
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    As soon as you have fixed the typo, you might be interested in the following question: http://math.stackexchange.com/q/306252/ – 2017-02-17
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    Yes! Thank you! – 2017-02-17

1 Answers 1

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Hope that it help you :to prove this $$E[\exp(\int_0^t\left(\frac{Bs}{1-s}\right)^2ds]<\infty$$ note that $0