# PDF for the integral of a Stochastic Process

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?

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There isn't enough information to find the pdf. –  Sasha May 4 '12 at 0:40
There isn't? Really? $P$ is known: I have at my disposal a PDF for the location of the process at any time $t$. I can't think of what more information would be needed. –  gmb May 4 '12 at 0:53
I suggest you improve your question to make it explicit what is known. For example, $P$ might be defined in terms of SDE, but the expression for pdf of $P(t)$ might not be available. In order to find pdf of $P(t)$, you need to find the joint pdf for $(P(t), P(t_f))$ and then condition on $P(t_f)=b$. –  Sasha May 4 '12 at 0:59

## 1 Answer

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 ?

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