Telegraph processes with random velocities

Stadje, Wolfgang; Zacks, Shelemyahu

doi:10.1017/s0021900200020465

Cited by 56 publications

(66 citation statements)

References 19 publications

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“…This is tantamount to saying that the Fourier transform h ν (ξ, t) is the solution to 18) and this completes the proof. 19) and let l ν (x, t) be the law of L ν (t), t > 0. We have the following result.…”

Section: 2mentioning

confidence: 99%

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Time-Changed Processes Governed by Space-Time Fractional Telegraph Equations

D’Ovidio

Orsingher

Toaldo

2014

Stochastic Analysis and Applications

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Abstract. In this work we construct compositions of vector processes of the form S 2β n c 2 L ν (t) , t > 0, ν ∈ 0, 1 2 , β ∈ (0, 1], n ∈ N, whose distribution is related to space-time fractional n-dimensional telegraph equations. We present within a unifying framework the pde connections of n-dimensional isotropic stable processes S 2β n whose random time is represented by the inverse L ν (t), t > 0, of the superposition of independent positively-skewed stable processes,, independent stable subordinators). As special cases for n = 1, ν = ) and the two-dimensional motion at finite velocity with a random time are investigated. For all these processes we present their counterparts as Brownian motion at delayed stabledistributed time.

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Section: 2mentioning

confidence: 99%

“…V (0) (with values ±c). Properties of this process (including first-passage time distributions) are studied in Foong and Kanno [8] and a telegraph process with random velocities has been recently considered by Stadje and Zacks [19]. For n = 1, β = 1 and ν = 1 2 the special equation…”

mentioning

confidence: 99%

Time-Changed Processes Governed by Space-Time Fractional Telegraph Equations

D’Ovidio

Orsingher

Toaldo

2014

Stochastic Analysis and Applications

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“…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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“…Many authors proposed suitable generalizations of the telegraph process, such as the 1-dimensional cases with three cyclical velocities [18], or with n values of the velocity [12], or with random velocities [21]. See also the paper by Lachal [14], where the cyclic random motion in R d with n directions is studied.…”

Section: Introductionmentioning

confidence: 99%