Telegraph Process on the Line

Kolesnik, Alexander D.; Ratanov, Nikita

doi:10.1007/978-3-642-40526-6_2

Cited by 2 publications

(3 citation statements)

References 20 publications

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“…Since (1.1) appears in electrical wave propagation models, it is known as the telegraph equation or the hyperbolic heat equation. An explicit solution of (1.1) can be carried out in terms of special functions, see for instance Chapter 2 in [29].…”

Section: Introductionmentioning

confidence: 99%

“…The telegraph process has been proposed as an alternative to diffusion models and, as such, extensively studied by the probability and physics communities. Its generalisations are ubiquitous in applications, including transport phenomena in physical and biological systems, and it has produced a vast mathematics literature; standard references include [5,12,14,15,19,20,21,22,23,24,26,27,28,29,30,37,40,46] and further references may be found from therein. It is also used in the context of risk theory and to model financial markets, see [16], [29] and [38].…”

Section: Introductionmentioning

confidence: 99%

“…Its generalisations are ubiquitous in applications, including transport phenomena in physical and biological systems, and it has produced a vast mathematics literature; standard references include [5,12,14,15,19,20,21,22,23,24,26,27,28,29,30,37,40,46] and further references may be found from therein. It is also used in the context of risk theory and to model financial markets, see [16], [29] and [38]. Statistics for the telegraph process are done in [22].…”

Section: Introductionmentioning

confidence: 99%

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Quantitative control of Wasserstein distance between Brownian motion and the Goldstein--Kac telegraph process

Barrera¹,

Lukkarinen²

2022

Preprint

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In this manuscript, we provide a non-asymptotic process level control between the telegraph process and the Brownian motion with suitable diffusivity constant via a Wasserstein distance with quadratic average cost. In addition, we derive non-asymptotic estimates for the corresponding time average p-th moments. The proof relies on coupling techniques such as coinflip coupling, synchronous coupling and the Komlós-Major-Tusnády coupling.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Quantitative control of Wasserstein distance between Brownian motion and the Goldstein--Kac telegraph process

Barrera¹,

Lukkarinen²

2022

Preprint

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NuZZ: Numerical Zig-Zag for general models

Pagani,

Chevallier,

Power

et al. 2024

Stat Comput

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Markov chain Monte Carlo (MCMC) is a key algorithm in computational statistics, and as datasets grow larger and models grow more complex, many popular MCMC algorithms become too computationally expensive to be practical. Recent progress has been made on this problem through development of MCMC algorithms based on Piecewise Deterministic Markov Processes (PDMPs), irreversible processes which can be engineered to converge at a rate which is independent of the size of the dataset. While there has understandably been a surge of theoretical studies following these results, PDMPs have so far only been implemented for models where certain gradients can be bounded in closed form, which is not possible in many relevant statistical problems. Furthermore, there has been substantionally less focus on practical implementation, or the efficiency of PDMP dynamics in exploring challenging densities. Focusing on the Zig-Zag process, we present the Numerical Zig-Zag (NuZZ) algorithm, which is applicable to general statistical models without the need for bounds on the gradient of the log posterior. This allows us to perform numerical experiments on: (i) how the Zig-Zag dynamics behaves on some test problems with common challenging features; and (ii) how the error between the target and sampled distributions evolves as a function of computational effort for different MCMC algorithms including NuZZ. Moreover, due to the specifics of the NuZZ algorithms, we are able to give an explicit bound on the Wasserstein distance between the exact posterior and its numerically perturbed counterpart in terms of the user-specified numerical tolerances of NuZZ.

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Telegraph Process on the Line

Cited by 2 publications

References 20 publications

Quantitative control of Wasserstein distance between Brownian motion and the Goldstein--Kac telegraph process

Quantitative control of Wasserstein distance between Brownian motion and the Goldstein--Kac telegraph process

NuZZ: Numerical Zig-Zag for general models

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