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Given the SDE: $dX_{t}=\sqrt{X_{t}}dW_{t},$ $\ X_{0}=1$ , where $W_{t}$ is a 1-d Brownian motion.

I was told that this SDE has a unique strong solution, but I don't know how to construct it. I know that this SDE has strong uniqueness, therefore I only need to construct a weak solution. I'm guessing we need to first consider a weak solution up to an explosion time (i.e. the hitting time of $X_{t}$ to the level $0$), but how to show such a weak solution exists?

Thank you for your help!

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Check this: $X_t = (\frac{1}{2}(W_t-W_0)+1)^2$ –  nikita2 Oct 19 '12 at 4:21
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@nikita2 This would satisfy $\mathrm{d}X_t = \frac{1}{4} \mathrm{d}t + \sqrt{X_t} \mathrm{d}W_t$. –  Sasha Oct 19 '12 at 4:39
    
@Sasha Ok I just thought about PDE with separated variables. –  nikita2 Oct 19 '12 at 4:43
    
Wouldn't a solution to this SDE be negative for some $t$ with positive probability? And isn't that a problem when looking at $\sqrt{X_t}$? –  Stefan Hansen Oct 22 '12 at 11:54
    
How do you know a strong solution exists? $\sqrt{x}$ is not Lipschitz at $0$. –  Michael Jul 12 at 11:19

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