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In general, we use Ito's formula to solve linear stochastic differential equations. Consider for instance the geometric brownian motion:

$$dX_t = \alpha X_t dt + \beta X_t dW.$$

My question is:

How can I solve this a non-linear stochastic differential equation, like:

$$dX_t = \alpha X_t dt + \beta \sqrt{X_t} dW.$$

Can I still use Ito's lemma or do I need to transform the SDE? If I have to transform the SDE, then how and what is the general rule? Any help is appreciated.

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  • $\begingroup$ I believe you meant $dX_t$ on your LHS. $\endgroup$
    – Vim
    Dec 15, 2018 at 14:31

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For most SDE, I don't think there exists "general rule" that can apply in finding their analytic solutions. For some explicitly solvable cases you may be interested in this.

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  • $\begingroup$ Thanks for your answer, but non of these case you are refering to is gonna solve my problem. $\endgroup$
    – gariban17
    Dec 15, 2018 at 15:04

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