Telegraph Processes and Option Pricing

Kolesnik, Alexander D.; Ratanov, Nikita

doi:10.1007/978-3-642-40526-6

Cited by 83 publications

(103 citation statements)

References 13 publications

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“…It should be noted that this result was previously obtained using other methods (see Refs. [25,33,34]). We are also concerned with the time dependence of the relaxation function µ(t).…”

Section: General Resultsmentioning

confidence: 99%

“…The dichotomous process is the simplest random process that plays an important role in numerous applications. If the probability densities p ± (τ ) are exponential, then many of the properties of this process can be determined by using the telegraph [23,24,25]. Since we are interested in studying the influence of different waiting time distributions on the character of relaxation of two-state systems, for this purpose it is more convenient to use the CTRW approach [26,27,28,29].…”

Section: Model Descriptionmentioning

confidence: 99%

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Continuous-time Random Walk Model of Relaxation of Two-state Systems

Denisov¹,

Bystrik²

2015

Acta Phys. Pol. B

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Using the continuous-time random walk (CTRW) approach, we study the phenomenon of relaxation of two-state systems whose elements evolve according to a dichotomous process. Two characteristics of relaxation, the probability density function of the waiting times difference and the relaxation law, are of our particular interest. For systems characterized by Erlang distributions of waiting times, we consider different regimes of relaxation and show that, under certain conditions, the relaxation process can be non-monotonic. By studying the asymptotic behavior of the relaxation process, we demonstrate that heavy and superheavy tails of waiting time distributions correspond to slow and superslow relaxation, respectively.

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“…It should be noted that this result was previously obtained using other methods (see Refs. [25,33,34]). We are also concerned with the time dependence of the relaxation function µ(t).…”

Section: General Resultsmentioning

confidence: 99%

Section: Model Descriptionmentioning

confidence: 99%

Continuous-time Random Walk Model of Relaxation of Two-state Systems

Denisov¹,

Bystrik²

2015

Acta Phys. Pol. B

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“…Other important properties of the telegraph random processes can be found in the recently published book [17].…”

Section: Some Basic Properties Of the Telegraph Processmentioning

confidence: 99%

Probability Distribution Function for the Euclidean Distance Between Two Telegraph Processes

Kolesnik

2014

Adv. Appl. Probab.

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Consider two independent Goldstein-Kac telegraph processes X 1 (t) and X 2 (t) on the real line R. The processes X k (t), k = 1, 2, describe stochastic motions at finite constant velocities c 1 > 0 and c 2 > 0 that start at the initial time instant t = 0 from the origin of R and are controlled by two independent homogeneous Poisson processes of rates λ 1 > 0 and λ 2 > 0, respectively. We obtain a closed-form expression for the probability distribution function of the Euclidean distance ρ(t) = |X 1 (t) − X 2 (t)|, t > 0, between these processes at an arbitrary time instant t > 0. Some numerical results are also presented.

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“…The explicit expressions for the density functions f i (x, t) of T i (t) follow from (2.4)-(2.6) by summing up in m, see, for example, [3], [13].…”

Section: Introductionmentioning

confidence: 99%

“…We start with the standard inhomogeneous integrated telegraph process T = T(t), based on the Markov process ε, see, for example, [13]. Let c 0 , c 1 ∈ R, c 0 > c 1 .…”

Section: Introductionmentioning

confidence: 99%

Double Telegraph Processes and Complete Market Models

Ratanov¹

2014

Stochastic Analysis and Applications

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The traditional jump-telegraph processes are based on a Poisson process with alternating intensities. We develop a new model based on an alternating doubly stochastic Poisson process with random intensities of jumps. Moreover, in this model the jump-telegraph process performs additional jumps each time when the intensity changes at random. Martingale measures for this type of processes are described by using Girsanov's transformation. The market model based on the doubly stochastic jump-telegraph process is studied. A class of complete models is also considered.

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Telegraph Processes and Option Pricing

Cited by 83 publications

References 13 publications

Continuous-time Random Walk Model of Relaxation of Two-state Systems

Continuous-time Random Walk Model of Relaxation of Two-state Systems

Probability Distribution Function for the Euclidean Distance Between Two Telegraph Processes

Double Telegraph Processes and Complete Market Models

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