I would like to find the probability density function (at stationarity) of the random variable $X_t$, where (I'm not sure this notation makes sense, I'm not very familiar with the stochastic calculus literature): \begin{equation*} dX_t = -aX_t + d N_t, \end{equation*} $a$ is a constant and $N_t$ is a compound Poisson process with Poisson jump size distribution.
In other words, $X_t$ solves the ordinary differential equation $\frac{d X_t}{dt} + a X_t=0$, but at times $t_i$ say, where the $t_i$ are exponentially distributed with mean $1/k$, $X_t$ increases by an integer drawn from $M\sim Poi(m)$ (i.e. $X_t$ gets a Poisson-distributed "kick" upwards at exponentially distributed intervals).
Is there a way of obtaining the pdf for this random variable $X$? If I have understood things correctly, the Kramers-Moyal equation for the pdf of $X$ is of infinite order because it is a jump Markov process. I have also tried looking at the Master Equation but I get lost. However, I am new to this literature and was wondering if the solution is easy for those in the know, since it is such a simple system.
Many thanks for your help!