Let $X_t, t\geq 0$ be a Poisson process with rate parameter $\lambda$. Compute the karhunen-loeve expansion of $X$ in interval [0, T]. How about the KL expansion of the centered process $X_t-\lambda t$?
The auto-correlation function of poisson process is $R(s,t)=\lambda^2 st + \lambda \min(s,t)$.
By definition, KL expansion should satisfy $\int_0^T R(s,t) \phi_n(t) dt = \lambda_n \phi_n(s)$.
I've problems figuring out how to solve the integrated equation.
For wiener process, this link (Karhunen–Loève approximation of Brownian motion and diffusions) and wikipedia article on KL expansion was useful.
Best Answer
The integral equation is solved as it is in the case of brownian motion and brownian bridge. The eigenfunctions are sine functions, and the tricky parts are the eigenvalues and the distribution of the random coefficients in the K-L expansion.
If g is the eigenfunction with eigenvalue \gamma, then \gamma*g = -lambda*g and g(0) = 0. If you substitute back into the integral equation for g then you get an equation for \gamma in terms of sine, cosine, and \lambda.
I have recently submitted a manuscript for publication giving the complete solution for the eigenfunction/eigenvalue part of this problem and some generalizations.
Prof Eric Key Dept of Math Sci UW-Milwaukee.