Estimation for the discretely observed telegraph process

Abstract

Abstract. The telegraph process {X(t), t > 0} is supposed to be observed at n + 1 equidistant time points t i = i∆ n , i = 0, 1, . . . , n. The unknown value of λ, the underlying rate of the Poisson process, is a parameter to be estimated. The asymptotic framework considered is the following: ∆ n → 0, n∆ n = T → ∞ as n → ∞. We show that previously proposed moment type estimators are consistent and asymptotically normal but not efficient. We study further an approximated moment type estimator which is still not efficient but comes in explicit form. For this estimator the additional assumption n∆ 3 n → 0 is required in order to obtain asymptotic normality. Finally, we propose a new estimator which is consistent, asymptotically normal and asymptotically efficient under no additional hypotheses.

On the Asymmetric Telegraph Processes

“…The moments of the telegraph processes in the homogeneous symmetric setting have been described by Iacus and Yoshida, [12], in terms of modified Bessel functions; see also [14]. Our approach produces expressions that are a bit more simple.…”

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

“…Properties of moment type estimators under the additional assumption n 3 n → 0 and in the asymptotic framework n n = T → ∞ have been studied in Iacus and Yoshida (2006). Under the current scheme, i.e.…”

“…The second paper is about the estimation of the parameter θ of the non-constant rate λ θ (t) from continuous observations of the process. Very recently, Iacus and Yoshida (2006) consider 3 estimation problem for this process under the large sample asymptotic scheme, i.e. T = n n → ∞ and n 3 n → 0 as n → ∞, but their approach is based on different arguments.…”

Linear combinations of the telegraph random processes driven by partial differential equations

“…The classical Goldstein-Kac telegraph process, first introduced in the works [10] and [14], describes the stochastic motion of a particle that moves at constant finite speed on the real line R and alternates two possible directions of motion at random Poisson time instants. The main properties of this process and its numerous generalizations, as well as some their applications, have been studied in a series of works [1][2][3][4][5][6][7][8][9], [11][12][13], [15,16], [19,20], [22][23][24][25][26][27][28][29], [31]. An introduction to the contemporary theory of the telegraph processes and their applications in financial modelling can be found in the recently published book [22].…”