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Subordinated compound Poisson processes of order k

Ayushi Singh Sengar Neelesh S. Upadhye

https://doi.org/10.15559/20-VMSTA165

Pub. online: 5 Nov 2020 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 7, Issue 4 (2020), pp. 395–413

Abstract

In this article, the compound Poisson process of order k (CPPoK) is introduced and its properties are discussed. Further, using mixture of tempered stable subordinators (MTSS) and its right continuous inverse, the two subordinated CPPoK with various distributional properties are studied. It is also shown that the space and tempered space fractional versions of CPPoK and PPoK can be obtained, which generalize the process defined in [Statist. Probab. Lett. 82 (2012), 852–858].

On shortfall risk minimization for game options

Yan Dolinsky

https://doi.org/10.15559/20-VMSTA164

Pub. online: 29 Oct 2020 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 7, Issue 4 (2020), pp. 379–394

Abstract

In this paper we study the existence of an optimal hedging strategy for the shortfall risk measure in the game options setup. We consider the continuous time Black–Scholes (BS) model. Our first result says that in the case where the game contingent claim (GCC) can be exercised only on a finite set of times, there exists an optimal strategy. Our second and main result is an example which demonstrates that for the case where the GCC can be stopped on the whole time interval, optimal portfolio strategies need not always exist.

Geometric branching reproduction Markov processes

Assen Tchorbadjieff Penka Mayster

https://doi.org/10.15559/20-VMSTA163

Pub. online: 30 Sep 2020 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 7, Issue 4 (2020), pp. 357–378

Abstract

We present a model of a continuous-time Markov branching process with the infinitesimal generating function defined by the geometric probability distribution. It is proved that the solution of the backward Kolmogorov equation is expressed by the composition of special functions – Wright function in the subcritical case and Lambert-W function in the critical case. We found the explicit form of conditional limit distribution in the subcritical branching reproduction. In the critical case, the extinction probability and probability mass function are expressed as a series containing Bell polynomial, Stirling numbers, and Lah numbers.

Ergodic properties of the solution to a fractional stochastic heat equation, with an application to diffusion parameter estimation

Diana Avetisian Kostiantyn Ralchenko

https://doi.org/10.15559/20-VMSTA162

Pub. online: 18 Sep 2020 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 7, Issue 3 (2020), pp. 339–356

Abstract

The paper deals with a stochastic heat equation driven by an additive fractional Brownian space-only noise. We prove that a solution to this equation is a stationary and ergodic Gaussian process. These results enable us to construct a strongly consistent estimator of the diffusion parameter.

On tail behaviour of stationary second-order Galton–Watson processes with immigration

Mátyás Barczy

Zsuzsanna Bősze Gyula Pap

https://doi.org/10.15559/20-VMSTA161

Pub. online: 10 Sep 2020 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 7, Issue 3 (2020), pp. 315–338

Abstract

Sufficient conditions are presented on the offspring and immigration distributions of a second-order Galton–Watson process ${({X_{n}})_{n\geqslant -1}}$ with immigration, under which the distribution of the initial values $({X_{0}},{X_{-1}})$ can be uniquely chosen such that the process becomes strongly stationary and the common distribution of ${X_{n}}$, $n\geqslant -1$, is regularly varying.

On distributions of exponential functionals of the processes with independent increments

Lioudmila Vostrikova

https://doi.org/10.15559/20-VMSTA159

Pub. online: 8 Sep 2020 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 7, Issue 3 (2020), pp. 291–313

Abstract

The aim of this paper is to study the laws of exponential functionals of the processes $X={({X_{s}})_{s\ge 0}}$ with independent increments, namely

\[ {I_{t}}={\int _{0}^{t}}\exp (-{X_{s}})ds,\hspace{0.1667em}\hspace{0.1667em}t\ge 0,\]

and also

\[ {I_{\infty }}={\int _{0}^{\infty }}\exp (-{X_{s}})ds.\]

Under suitable conditions, the integro-differential equations for the density of ${I_{t}}$ and ${I_{\infty }}$ are derived. Sufficient conditions are derived for the existence of a smooth density of the laws of these functionals with respect to the Lebesgue measure. In the particular case of Lévy processes these equations can be simplified and, in a number of cases, solved explicitly.

On infinite divisibility of a class of two-dimensional vectors in the second Wiener chaos

Andreas Basse-O’Connor Jan Pedersen Victor Rohde

https://doi.org/10.15559/20-VMSTA160

Pub. online: 28 Aug 2020 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 7, Issue 3 (2020), pp. 267–289

Abstract

Infinite divisibility of a class of two-dimensional vectors with components in the second Wiener chaos is studied. Necessary and sufficient conditions for infinite divisibility are presented as well as more easily verifiable sufficient conditions. The case where both components consist of a sum of two Gaussian squares is treated in more depth, and it is conjectured that such vectors are infinitely divisible.

Simple approximations for the ruin probability in the risk model with stochastic premiums and a constant dividend strategy

Olena Ragulina

https://doi.org/10.15559/20-VMSTA157

Pub. online: 4 Aug 2020 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 7, Issue 3 (2020), pp. 245–265

Abstract

We deal with a generalization of the risk model with stochastic premiums where dividends are paid according to a constant dividend strategy and consider heuristic approximations for the ruin probability. To be more precise, we construct five- and three-moment analogues to the De Vylder approximation. To this end, we obtain an explicit formula for the ruin probability in the case of exponentially distributed premium and claim sizes. Finally, we analyze the accuracy of the approximations for some typical distributions of premium and claim sizes using statistical estimates obtained by the Monte Carlo methods.

Approximations of the ruin probability in a discrete time risk model

David J. Santana Luis Rincón

https://doi.org/10.15559/20-VMSTA158

Pub. online: 4 Aug 2020 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 7, Issue 3 (2020), pp. 221–243

Abstract

Based on a discrete version of the Pollaczeck–Khinchine formula, a general method to calculate the ultimate ruin probability in the Gerber–Dickson risk model is provided when claims follow a negative binomial mixture distribution. The result is then extended for claims with a mixed Poisson distribution. The formula obtained allows for some approximation procedures. Several examples are provided along with the numerical evidence of the accuracy of the approximations.

Distance from fractional Brownian motion with associated Hurst index

0 < H < 1 / 2

to the subspaces of Gaussian martingales involving power integrands with an arbitrary positive exponent

Oksana Banna

Filipp Buryak Yuliya Mishura

https://doi.org/10.15559/20-VMSTA156

Pub. online: 23 Jun 2020 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 7, Issue 2 (2020), pp. 191–202

Abstract

We find the best approximation of the fractional Brownian motion with the Hurst index $H\in (0,1/2)$ by Gaussian martingales of the form ${\textstyle\int _{0}^{t}}{s^{\gamma }}d{W_{s}}$, where W is a Wiener process, $\gamma >0$.

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