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I would like to solve a SDE similar to the GBM but with an additional term. Generally, the process I'm looking for should have the following recursive form:
$$ \frac{dX_t}{X_t} = \alpha dt + \gamma dW_t - \phi X_t dt, $$ $$ d X_{t} = \alpha X_t dt + \gamma X_t dW_t - \phi X_t^2 dt. $$

I have attempted solving the SDE without success using Ito's Lemma when $f(X_t) = ln(X_t) $
$$ d ln(X_t) = \frac{1}{X_t} dX_t - \frac{1}{2}\frac{1}{X_t^2} (dX_t)^2 $$ $$ d ln(X_t) = \frac{dX_t}{X_t} - \frac{1}{2X_t^2} (\gamma X_t dW_t)^2 $$ $$ d ln(X_t) = \frac{d X_t}{X_t} - \frac{1}{2} \gamma^2 dt $$ $$ \frac{d X_t}{X_t} = d ln(X_t) + \frac{1}{2} \gamma^2 dt $$

and when $ f(X_t) = X_t^{-1} $ (EDIT: fixed some mistakes below) $$ d (X_t^{-1}) = - \frac{1}{X_t^2} dX_t + \frac{1}{2}\frac{2}{X_t^3} (dX_t)^2 $$ $$ d (X_t^{-1}) = - \frac{1}{X_t^2} (\alpha X_t dt + \gamma X_t dW_t - \phi X_t^2 dt) + \frac{1}{X_t^3} (\gamma X_t dW_t)^2 $$ $$ d (X_t^{-1}) = - \alpha X_t^{-1} dt - \gamma X_t^{-1} dW_t + \phi dt + \gamma^2 X_t^{-1} dt $$ $$ d f_t = \phi dt + (\gamma^2 - \alpha) f_t dt - \gamma f_t dW_t $$ $$ d f_t = (\phi + \gamma^2 f_t - \alpha f_t) dt - \gamma f_t dW_t $$

By substituting the result from the first attempt with Itô's Lemma into the original SDE I obtained $$ d ln(X_t) + \frac{1}{2} \gamma^2 dt = \alpha dt + \gamma dW_t - \phi X_t dt $$ $$ ln(X_t) - ln(X_0) = (\alpha - \frac{1}{2} \gamma^2) t + \gamma W_t - \phi \int_{0}^{t} X_t dt $$ $$ ln(X_t) + \phi \int_{0}^{t} X_t dt = ln(X_0) (\alpha - \frac{1}{2} \gamma^2)t + \gamma W_t. $$

EDIT: The result from the second attempt with Itô's Lemma might be used in the same way as here https://quant.stackexchange.com/questions/50485/solve-the-following-sde-mathrmdx-t-ab-x-t-mathrmdt-c-x-t-mat and maybe here How to solve the following SDE: $\mathrm{d}X_t = a(b-X_t) \mathrm{d}t + c X_t \mathrm{d}W_t$

EDIT: How should I proceed?

Techonomist
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  • You have a little mistake, note that $(dX_t)^2 = (\gamma X_t)^2dt$. – Math-fun Feb 19 '22 at 07:41
  • Thank you for the comment. Simplified that line but couldn't find the mistake. Is there a mistake still? – Techonomist Feb 19 '22 at 12:23
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    The equation you are considering has been extensively studied in population dynamics and systems ecology. It is a stochastic logistic equation or . With this you should be able to find what you are looking for; e.g. http://www.numdam.org/article/RSMUP_2001__106__165_0.pdf – KBS Feb 19 '22 at 13:10
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    here is another mistake: the Ito derivative for the second fucntion is incorrect; the second derivative should be multiplied by 1/2... I did not check more details – Math-fun Feb 22 '22 at 21:32
  • Thanks. I fixed the mistake and applied your answer. Now we only have different signs. – Techonomist Feb 24 '22 at 08:17
  • Great! I have editted the sign :-) – Math-fun Feb 25 '22 at 16:37

1 Answers1

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$\ln$ is not helping you here. Use $f(x)=x^{-1}$ to obtain with Ito's Lemma that $$df=-(\alpha f-\gamma^2 f-\phi)dt - \gamma f dW,$$ which is linear and admits a closed form solution. Once you obtain the solution, you have $x = \frac{1}{f}$.

Math-fun
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