# Has anyone found a way to determine the invariant measure of a one-dimensional jump-diffusion?

I am working with a jump-diffusion on the unit interval, with absorbing endpoints, and I was hoping someone has found a way to determine the invariant measure, similar to that of an Ito process.

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The general procedure is the following. Take the generator $L$ of the semigroup associated to your process. Then find its dual $L^*$. The latter governs the evolution of 1-dim distributions (via forward Kolmogorov equation), and an invariant density $p$ satisfies $L^*p=0$, should be positive and integrate to 1.

I suspect all this can be found in the Ethier&Kurtz book on Markov processes, but I do not have it at hand right now.

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Thank You! This concept really helped! –  user11484 Jan 26 '11 at 0:07
@unknown (google): A second look at your question makes me think that since the boundary is absorbing, there should be invariant measures supported by the boundary. –  Yuri Bakhtin Feb 2 '11 at 22:20
This is a very true statement, supported by an application of the stochastic Lyapunov method. Unfortunately, I cannot determine the probabilities of each of the boundary points. Thanks for thinking about my problem! –  user11484 Feb 2 '11 at 23:12
Indeed, any measure on the boundary is invariant in your case (and I should have thought about this when I was giving my initial answer). None of them is special, and in your question you probably did not give all details, since you seem to be looking for some specific measure. –  Yuri Bakhtin Feb 4 '11 at 2:08
I am so sorry. You're right, I am hoping to find or give a result similar to a theorem in Gihman and Skorohod for diffusions. The specific invariant measure given for each particular process is very important. Thanks! –  user11484 Feb 6 '11 at 22:10

Henry Tuckwell's 1976 paper, "On the First-Exit Time Problem for Temporally Homogeneous Markov Processes", gives an ODDE to solve to determine the invariant (ergodic) measure.

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