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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.