Parametric estimation for the standard and geometric telegraph process observed at discrete times

The telegraph process X(t), t ≥ 0, (Goldstein, Q J Mech Appl Math 4:129–156, 1951) and the geometric telegraph process $$S(t) = s_{0} {\rm exp}\{(\mu -\frac12\sigma^{2})t + \sigma X(t)\}$$ with μ a known real constant and σ > 0 a parameter are supposed to be observed at n + 1 equidistant time points t i = iΔ n ,i = 0,1,..., n. For both models λ, the underlying rate of the Poisson process, is a parameter to be estimated. In the geometric case, also σ > 0 has to be estimated. We propose different estimators of the parameters and we investigate their performance under the asymptotics, i.e. Δ n → 0, nΔ n = T < ∞ as n → ∞, with T > 0 fixed. The process X(t) in non markovian, non stationary and not ergodic thus we build a contrast function to derive an estimator. Given the complexity of the equations involved only estimators on the first model can be studied analytically. Therefore, we run an extensive Monte Carlo analysis to study the performance of the proposed estimators also for small sample size n