Lectures on Random Evolution

“…the distribution of X(t), is a mixture of continuous and discrete components. This distribution has been obtained by Goldstein [8], Orsingher [17], and Pinsky [18] with different techniques. The transition density is given by …”

Section: Moments Of the Telegraph Processmentioning

confidence: 93%

“…Many authors analyzed probabilistic properties of the process over the years (see e.g. Orsingher [16] and [17]; Pinsky [18]; Fong and Kanno [7]; Stadje and Zacks [21]). Di Crescenzo and Pellerey [4] proposed the geometric telegraph process as a model to describe the dynamics of the price of risky assets, i.e.…”

Section: Introductionmentioning

confidence: 99%

Estimation for the discretely observed telegraph process

Iacus

Yoshida

2009

Theor. Probability and Math. Statist.

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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.

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“…Random dynamical systems [15,17] take into consideration some very important and widely studied cases, namely dynamical systems generated by learning systems [1,20,22,29], iterated function systems with an infinite family of transformations [37,38], Poisson driven stochastic differential equations [16,35,36], random evolutions [11,32] and irreducible Markov systems [41], used for the computer modelling of different stochastic processes.…”

Section: Introductionmentioning

confidence: 99%