I would like to find the probability density function (at stationarity) of the random variable $X_t$, where: \begin{equation*} dX_t = -aX_t dt + 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!
Addendum:
In the following paper, an expression for the Fourier transform of the probability density function is provided for general jump size distribution (see Section 5.2):
Generalized Fokker-Planck equation: Derivation and exact solutions
or in the ArXiv: http://arxiv.org/pdf/0808.0274.pdf
For the case I am interested in (Poisson jump size distribution), I don't think the Fourier transform can be inverted analytically. However, the paper gives the exact solution for exponentially-distributed jump sizes as an example.
Thank you to those who have answered or commented so far. It is actually very helpful for me to know that there isn't a straightforward way to obtain the pdf (if at all). Nevertheless, if anyone else out there does know of a way I'd be interested to hear about it.