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

Abstract

The telegraph process X(t), t ≥ 0, (Goldstein, Q J Mech Appl Math 4:129-156, 1951) and the geometric telegraph process S(t) = s0 exp{μ -1/2σ2)t + σ 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. © 2007 Springer Science+Business Media B.V

On a jump-telegraph process driven by an alternating fractional Poisson process

“…For the sake of brevity, we only mention two works: Ratanov [36], in which the author proposed a new generalisation of the jump-telegraph process with variable velocities and jumps, and then applied this construction to market modelling; and Kolesnik and Ratanov (see [21] and the references therein), in which the authors presented a thorough investigation on the telegraph process and its applications to option pricing. Estimation procedures for the standard and geometric telegraph process (see [5], [6], and [17]), and for a Brownian motion governed by a telegraph process [34] have been recently provided under the hypothesis of discrete-time sampling.…”

Least-squares change-point estimation for the telegraph process observed at discrete times

“…. , n, n∆ n = T and ∆ n → 0 as n → ∞, De Gregorio and Iacus (2006) proposed pseudo-maximum likelihood and implicit moment based estimators for the rate λ of the telegraph process. Under the additional condition n∆ n → ∞ as n → ∞, Iacus and Yoshida (2007) studied the asymptotic properties of explicit moment type estimators and further propose a consistent, asymptotically gaussian and asymptotically efficient estimator based on the increments of the process.…”

“…The change point estimation theory has been employed widely by means of the likelihood function (see Csörgő and Horváth, 1997). Unfortunately, the likelihood function for the telegraph process is not known and the pseudo likelihood proposed in De Gregorio and Iacus (2006) is not easy to treat in this framework. We will then proceed using the alternative method based on least squares proposed in Bai (1994Bai ( , 1997 and used in different contexts by many authors including Hsu (1977Hsu ( , 1979 for the i.i.d.…”

Parametric estimation for planar random flights

“…Iacus (2001) considers the estimation of the parameter θ of a non-constant rate λ θ (t). More recently, De Gregorio and Iacus (2008) introduced a pseudo-maximum likelihood estimator and a moment estimator for the parameter λ when the sample paths of the telegraph process are observed only at equidistant discrete times. The authors also analyze the same statistical problem for a geometric telegraph process particularly interesting in view of financial applications.…”