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$$dX_t = -\frac{1}{2}e^{-2X_t}\ \ dt+e^{-X_t}dB_t, X_0=x_0$$

Hint: solve this equation using the substitution $X_t=u(B_t)$, show that the solution blows up at a finite random time.

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    Can one have **less** input in a question?2012-06-09
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    I checked it again. It's the same as that on the exam paper.2012-06-10
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    So there **was** a hint, after all. By the way: what is blocking you here? What similar questions can you solve? What did you try? What are your thoughts?2012-06-10
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    I want to use Ito's lemma and then to apply the coefficient matching. But I get a strange result. I think there may be something wrong, so I just keep the oringinal problem here.2012-06-10
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    You might want to show your *strange result*.2012-06-10
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    OK. I add what I have done to the problem discreption.2012-06-10
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    Why $f(t)=1/(t-1)$?2012-06-10
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    Just solve the ODE $f'=-f^2$2012-06-10
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    Your $f$ is not the only solution of $f'=-f^2$, is it?2012-06-10
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    No. In fact this is another problem. I don't know how to get the analytical solution of the ODE $f'=f^2$.2012-06-10
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    I would say this is your **main** problem: to determine the solutions of the ODE $y'=-y^2$. Sure you cannot do this? Hint: the ODE means $-y'/y^2=1$.2012-06-10
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    $y(t)=\frac{1}{t+C}$2012-06-10
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    Right. Now you have everything that is needed to complete the answer.2012-06-10
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    Thanks very much. But I'm still confused about the meaning of "the solution blows up at a finite random time" mean? Does it mean that after finite random time, $B_t$ hits $−C$ , and $X_t$ goes to infinity?2012-06-10

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