My continuous-time, continuous step Stochastic Process P runs from time $t=0$ to $t=t_f$ and generates a path. I am able to observe its starting and ending position (so $P(0)=a$ and $P(t_f)=b$), but I'm unsure what happened in the middle. I want to come up with a PDF for the integral of the process from $0$ to $t_f$.
Any advice?
If I'm not mistaken, if you have a SDE for your process under Ito form, there is under conditions a way to go to the forward and backward equations which gives the time evolution of the probability density. Is it helpful ?