Hi,I was wondering whether there is a closed-form solution for the expectation of the inverse of an OU-process?X follows a standard OU-process: Any help appreciated.Best regardsobs

thanks a lot for your answerSo I guess if the OU process fluctuates around a non-zero mean does not change this fact?

Non-zero mean Gaussian is equal to mean + zero-mean Gaussian. However 1/X has infinite mean whatever the mean - direct calculation.

It exists as a Cauchy principal value integral. Example: eps f[eps]1/10 1.053491/100 1.066761/1000 1.068081/10000 1.068211/100000 1.068221/1000000 1.06822One-line version: 1.06822

Yes, Cauchy principal values can be calculated but they are not equal to - this expectation does not exist.

As a general rule, that's probably right. But, if you needed it to exist, well there you go ...

Not sure what you're asking about the known constant.If you needed E[1/X] to exist as a law of large numbers result, and wanted it to agree with the PV integral,I guess you'd have to make the sampling procedure consistent with the Cauchy PV defn. Whether or not that is possible, I'm not sure. Vaguely, given an infinite sample sequence,you could define an {eps,N}-sample average to mean one that took the average of the 1/X, wherethe data are the first N samples that lie in the set {X: |X| > eps}. Each {eps,N}-sample avg of 1/X should converge a.s. to a finite value as N -> infty,for example the f[eps] earlier. Then, define the final sample average to be the eps -> 0 limit of those. Does it really work and seem a reasonable definition ofhow to take an infinite limit sample average in this improper case? I'm not sure, but seems reasonable to me.

Yes, the example I posted had principal value = pi e^(-4) erfi(2), where erfi(z) = imag. error function = erf(i z)/i I agree that 'undefined' is a good general purpose answer. Sometimes, weird things turn up where undefined is not the right answer.A good example from physics are integrals in QED which formally are infinite, but which, after something of a handwave, yield a finite part which is exactly the right answer.

That is a coincidence! Besides brute force cut-offs, I really only recall one good trick for renormalization: 'dimensional regularization', where the diverging integrals in question are int f(x) dx, dx is a d-dimensional vol element. Then, by changing d to say d - epsilon, the integral convergences. So, different than Cauchy idea. But in physics generally, esp with some kind of wave propagation, poles show up in integrations all the time and they have to be moved somewhere, producing retarded and advanced waves,the latter esp. good for seeing into the future! p.s. I see we have some businesses emerging to exploit this