# Compound Poisson Process Expected Exit Time

It is very well known that expected time for a standard Brownian motion to exit from interval $[a,b]$ (where $a<0$ and $b>0$) is $-ab$. In one of my projects, I wanted to calculate the similar quantity for a compound poisson process. I am not an expert in handling point processes and would be grateful for any help.

Question : $X$ is a compound poisson process starting at position $0$ with arrival rate $\lambda$ and the distribution of jumps as $F(dz)$. What would be the expected time $\tau$ for $X$ to exit $[-a,a]$? Also, what would be the distribution of $X_\tau$?

• The method generally used to find the distribution of the first exit time of a Brownian motion $B(t)$ from $[-a,a]$ relies heavily on the fact that $B(t)$ is a martingale (and hence so too is $B(t)-t^2$). A general compound Poisson process is not so easily analyzed. – Math1000 Jun 14 '18 at 3:32
• Do you have a particular jump distribution in mind? Seem to recall exponentially distributed jumps were analytically tractable. Certainly mean and variance can be characterised at least in general but not sure about hitting times. Interesting problem though... – Mehness Jun 14 '18 at 7:25
• @Math1000: I guess you meant $B^2(t)-t$. Not sure how people tackle Poisson type of processes. – chandu1729 Jun 14 '18 at 11:42
• @Mehness : I would be first interested in Gaussian jump distribution. Do you think it would be tractable? What are the usual methods people use for these kind of problems? Thanks. – chandu1729 Jun 14 '18 at 11:43
• this thesis contains the following statement, for what it's worth: "Unfortunately, it seems to be impossible to determine the distribution of a crossing time for general L ́evy processes. The distributions of first hitting times are only known in some very special cases, such as L ́evy processes with bounded variation, see e.g. Gusak and Koralyuk (1968), or in case of a Brownian motion with constant drift, see e.g. Harrison (1985)" – Mehness Jun 14 '18 at 12:07