Computing a density function for the integral of a stochastic process, given its transition function - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T14:10:53Z http://mathoverflow.net/feeds/question/112102 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/112102/computing-a-density-function-for-the-integral-of-a-stochastic-process-given-its Computing a density function for the integral of a stochastic process, given its transition function GB 2012-11-11T18:34:48Z 2012-11-11T19:09:53Z <p>$P$ 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 $0 \le t_a &lt; t_b \le t_f$, 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).</p> <p>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 $x_0$?</p>