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The equation in question:

$$dX_t = r X_t (1-X_t) dt + \sigma X_t(1-X_t) dW_t $$

I won't bore you with the pages after pages of failed attempts. My best shot was considering $f(x,t) = e^{rt} (1-x)$

and cranking Ito' s lemma.

$$df(X_t) = - e^{rt} dX_t + re^{rt} (1-X_t) dt $$

Copious expansion and rearrangement leads to stuff like

$$df(X_t) = re^{rt}(X_t+1)(1-X_t) dt + \sigma e^{rt} X_t(1-X_t) dW_t $$

which gives

$$\frac{df(X_t)}{f(X_t)} = r(X_t+1) dt + \sigma X_t dW_t = rdt + X_t (r dt+ \sigma dW_t) $$

which looks pretty but I don't know how to solve.

(Considering $f(x,t)=e^{rt} X_t$ isn't much better. There's an obvious symmetry that comes out if you write this as a system of two equations, one for $dX_t$ and one for $dY_t = (1-X_t)$.)

Any ideas? This isn't even for work or school, I'm just obsessing about it.

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