The classical bi-Poisson process: an invertible quadratic harness (Q2507713)

The authors prove an elementary construction of a class of so-called bi-Poisson processes, i.e. a special class of square-integrable Markov processes with mean \(0\), \(\text{Cov}(X_t, X_s)=\min\{t, s\}\), and with certain given conditions for conditional means (the harness condition) and conditional variances. The main point is that the construction yields (a) an example of a time-invertible process without smooth transition probabilities, and (b) a non-trivial quadratic harness suitable for further analysis.