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Given $$f(x) = a_0 + a_1\,x + a_2\,x^2 + \dots + a_p\,x^p$$ $$g(x) = b_0 + b_1\,x + b_2\,x^2 + \dots + b_q\,x^q$$

Is there any conclusion about the strong solution for following SDE: $$dX_t = f(X_t)\,dt + g(X_t)\,dW_t$$

Most results I found require drift and diffusion coefficients are Lipschitz...

Thank you very much!

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    Initial thoughts are if you can arrange it so that $g(x) = b_0$, then let $F$ be the (negative of the) integral of $f(x)$ so that the SDE can be rewritten as $dX_t = - F^{\prime}(x) dt + b_0 dW_t$ so that if it exists, which will depend on the order of the polynomial $F$ then the SDE will have a stationary distribution proportional to $\exp(-U(x)/b_0)$, but otherwise that is a very general class of diffusions – 2017-02-19
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    Hi Nadiels, could you please elaborate more about the approach? Is there any reference resources? Thank you very much. – 2017-09-27

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