Stratonovich SDE and main hint for Ito's lemma

Question

I've stuck on a SDE problem. Namely, I've got to solve equation $$ \text{d}X_t = e^{-X_t}\circ\text{d}B_t. $$ So in order to apply Ito's lemma, I transformed this SDE into form $$ \text{d}X_t = -\frac{1}{2}e^{-2X_t}\text{d}t + e^{-X_t}\text{d}B_t. $$ I don't have any clue how to pick the proper function to apply Ito's lemma on. Is there any some sort of scheme of picking such functions on various SDEs? Moreover, I would love to get some hints about the literature with some of SDEs examples, solving techniques and applications.

Answer 1

For $Y_{t}=e^{2X_{t}}$ we get

$$dY_{t}=Y_{t}dB_{t}$$

and so as mentioned Solution to General Linear SDE, we get

$$Y_{t}=Y_0\exp\left( \int_0^t\left(- \frac{1}{2}s^{2} \right) \mathrm{d}s + \int_0^t s\mathrm{d}B_s\right).$$

I picked a function that would lead to cancelations i.e. $e^{2X_{t}}e^{-2X_{t}}=1$.