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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$2012-10-19
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    @nikita2 This would satisfy $\mathrm{d}X_t = \frac{1}{4} \mathrm{d}t + \sqrt{X_t} \mathrm{d}W_t$.2012-10-19
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    @Sasha Ok I just thought about PDE with separated variables.2012-10-19
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    Wouldn't a solution to this SDE be negative for some $t$ with positive probability? And isn't that a problem when looking at $\sqrt{X_t}$?2012-10-22
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    How do you know a strong solution exists? $\sqrt{x}$ is not Lipschitz at $0$.2014-07-12
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    @Michael You only need Holder-$1/2$ for the diffusive term in 1D SDE's. See for instance Section 3 in chapter IX of Revuz and Yor.2017-08-15

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