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I have a continuous-time stochastic process $X$, described as follows:

(1) If the process is at $x_0$ at time $t_0$, then the function $f(t_f, x_f \, | \, t_0, x_0)$ is a PDF in the parameter $x_f$ that describes the odds of the process being at $x_f$ at time $t_f$.

I want to find a PDF for the Ito Integral of a realization of this process. My understanding is that the first step towards this goal is to rephrase the process as a stochastic differential equation (please correct me if this is untrue!). A stochastic differential equation looks like:

(2) $X(0) = k$ and $dX(t) = \mu(X(t), t)dt + \sigma(X(t), t) dB(t)$

Where $k$ is some constant (which I know) and $B(t)$ is a Brownian Motion representing the random variation in my process.

So my question is: given $f$ as described in (1), find $\mu$ and $\sigma$ as described in (2).

Or, alternately, tell me that I am being silly and that there is an easier way to Ito-integrate my process.

Thanks for your help!

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It is possible to find f as described in (1), given μ and σ as described in (2), by solving the Kolmogorov forward equation. But given f, you have to ask yourself a number of questions. Is X(t) a continuous semi-martingale as described in (2)? It could be a jump process or even a jump-diffusion. Once you know for sure that X(t) looks like (2), you might have to reverse engineer the process of solving the Kolmogorov forward equation for f, given given μ and σ. It might also be possible to use some filtering process to determine μ and σ.

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