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$P$ is a one-dimensional Markov stochastic process that runs on time interval $[0, t_f]$. I know its transition function: $P(0) = x_0$ and for any $0 \le t_a < t_b \le t_f$, the function $f(x_b | x_a, t_a, t_b)$ describes the probability that $P(t_b) = x_b$ given that $P(t_a) = x_a$ (so $f$ is a density function in its first parameter).

Now, let $I$ be the random variable described by $\int_0^{t_f} P(s) ds$ for a random realization of $P$. Is it possible to find a density function for $I$ in terms of $f$ and $x_0$?

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# Computing a density function for the integral of a stochastic process, given its transition function

$P$ is a one-dimensional stochastic process that runs on time interval $[0, t_f]$. I know its transition function: $P(0) = x_0$ and for any $0 \le t_a < t_b \le t_f$, the function $f(x_b | x_a, t_a, t_b)$ describes the probability that $P(t_b) = x_b$ given that $P(t_a) = x_a$ (so $f$ is a density function in its first parameter).

Now, let $I$ be the random variable described by $\int_0^{t_f} P(s) ds$ for a random realization of $P$. Is it possible to find a density function for $I$ in terms of $f$ and $x_0$?