Solve SDE (Solved)

Question

Consider logistic growth with multiplicative noise, that is, the SDE $$dX_t = (\lambda X_t-X_t^2)dt + \sigma X_t dB_t$$, where $\lambda > 0$ and $\sigma > 0$ are fixed constants.

Answer 1

Applying Ito's lemma to $Y_t=X_t^{-1}$ gives

$$dY_t=-X_t^{-2}dX_t+X_t^{-3}\sigma^2X_t^2dt$$

$$=-\lambda X_t^{-1}dt+dt-\sigma X_t^{-1}dB_t+ \sigma^2X_t^{-1}dt$$

$$=(1-(\lambda+\sigma^2))X_t^{-1}dt-\sigma X_t^{-1}dB_t$$

$$dY_t=(1-(\lambda+\sigma^2)Y_t)dt-\sigma Y_tdB_t$$

Now the rest follows as in here: https://quant.stackexchange.com/questions/50485/solve-the-following-sde-mathrmdx-t-ab-x-t-mathrmdt-c-x-t-mat.