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Pathwise asymptotics for Volterra processes conditioned to a noisy version of the Brownian motion

Barbara Pacchiarotti

https://doi.org/10.15559/20-VMSTA149

Pub. online: 27 Feb 2020 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 7, Issue 1 (2020), pp. 17–41

Abstract

In this paper we investigate a problem of large deviations for continuous Volterra processes under the influence of model disturbances. More precisely, we study the behavior, in the near future after T, of a Volterra process driven by a Brownian motion in a case where the Brownian motion is not directly observable, but only a noisy version is observed or some linear functionals of the noisy version are observed. Some examples are discussed in both cases.

A characterization of equivalent martingale measures in a renewal risk model with applications to premium calculation principles

Nikolaos D. Macheras Spyridon M. Tzaninis

https://doi.org/10.15559/20-VMSTA148

Pub. online: 20 Feb 2020 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 7, Issue 1 (2020), pp. 43–60

Abstract

Generalizing earlier work of Delbaen and Haezendonck for given compound renewal process S under a probability measure P we characterize all probability measures Q on the domain of P such that Q and P are progressively equivalent and S remains a compound renewal process under Q. As a consequence, we prove that any compound renewal process can be converted into a compound Poisson process through a change of measures and we show how this approach is related to premium calculation principles.

The laws of iterated and triple logarithms for extreme values of regenerative processes

Alexander Marynych Ivan Matsak

https://doi.org/10.15559/20-VMSTA147

Pub. online: 17 Feb 2020 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 7, Issue 1 (2020), pp. 61–78

Abstract

We analyze almost sure asymptotic behavior of extreme values of a regenerative process. We show that under certain conditions a properly centered and normalized running maximum of a regenerative process satisfies a law of the iterated logarithm for the lim sup and a law of the triple logarithm for the lim inf. This complements a previously known result of Glasserman and Kou [Ann. Appl. Probab. 5(2) (1995), 424–445]. We apply our results to several queuing systems and a birth and death process.

Estimates for distribution of suprema of solutions to higher-order partial differential equations with random initial conditions

Yuriy Kozachenko Enzo Orsingher Lyudmyla Sakhno Olga Vasylyk

https://doi.org/10.15559/19-VMSTA146

Pub. online: 17 Dec 2019 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 7, Issue 1 (2020), pp. 79–96

Abstract

In the paper we consider higher-order partial differential equations from the class of linear dispersive equations. We investigate solutions to these equations subject to random initial conditions given by harmonizable φ-sub-Gaussian processes. The main results are the bounds for the distributions of the suprema for solutions. We present the examples of processes for which the assumptions of the general result are verified and bounds are written in the explicit form. The main result is also specified for the case of Gaussian initial condition.

Subject index

https://doi.org/10.15559/19-VMSTA64SI

Pub. online: 12 Dec 2019 Type: Subject Index

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 6, Issue 4 (2019), pp. 523–527

2010 Mathematics Subject Classification index

https://doi.org/10.15559/19-VMSTA64MI

Pub. online: 12 Dec 2019 Type: 2010 Mathematics Subject Classification Index

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 6, Issue 4 (2019), pp. 519–522

Keywords index

https://doi.org/10.15559/19-VMSTA64KI

Pub. online: 12 Dec 2019 Type: Keywords Index

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 6, Issue 4 (2019), pp. 517–518

Author index

https://doi.org/10.15559/19-VMSTA64AI

Pub. online: 12 Dec 2019 Type: Author Index

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 6, Issue 4 (2019), p. 515

Jackknife covariance matrix estimation for observations from mixture

Rostyslav Maiboroda Olena Sugakova

https://doi.org/10.15559/19-VMSTA145

Pub. online: 7 Nov 2019 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 6, Issue 4 (2019), pp. 495–513

Abstract

A general jackknife estimator for the asymptotic covariance of moment estimators is considered in the case when the sample is taken from a mixture with varying concentrations of components. Consistency of the estimator is demonstrated. A fast algorithm for its calculation is described. The estimator is applied to construction of confidence sets for regression parameters in the linear regression with errors in variables. An application to sociological data analysis is considered.

BSDEs and log-utility maximization for Lévy processes

Paolo Di Tella Hans-Jürgen Engelbert

https://doi.org/10.15559/19-VMSTA144

Pub. online: 28 Oct 2019 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 6, Issue 4 (2019), pp. 479–494

Abstract

In this paper we establish the existence and the uniqueness of the solution of a special class of BSDEs for Lévy processes in the case of a Lipschitz generator of sublinear growth. We then study a related problem of logarithmic utility maximization of the terminal wealth in the filtration generated by an arbitrary Lévy process.

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Online ISSN: 2351-6054
Print ISSN: 2351-6046

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