I have some struggles regarding solving SDEs. I know how the machinery works and all that, I just don't know how to make an ansatz when starting to solve a SDE.
For example when solving the Langevin SDE it is used that the starting solution/ansatz is $f(x,t)=x(t)e^{-ct}$ for the SDE (Langevin) given by $_=_+$.
So I wonder, How do I come up with such an ansatz $f(x,t)$. I have searched all the internet and I don't find it anywhere. My exam is getting closer and I hope someone can help me. I would really appreciate this.
I want to learn a systematic way of finding these kind of ansatz. Because the SDE one want to solve can be arbitrary, and hence I cannot memorise different kind of ansatz.
I really hope someone can help me. I just getting answers like "it is just like that" etc.