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2010 Mathematics Subject Classification index

https://doi.org/10.15559/23-VMSTA104MI

Pub. online: 13 Oct 2023 Type: 2010 Mathematics Subject Classification Index

Journal: Modern Stochastics: Theory and Applications Volume 10, Issue 4 (2023), pp. 463–466

Keywords index

https://doi.org/10.15559/23-VMSTA104KI

Pub. online: 13 Oct 2023 Type: Keywords Index

Journal: Modern Stochastics: Theory and Applications Volume 10, Issue 4 (2023), pp. 461–462

Author index

https://doi.org/10.15559/23-VMSTA104AI

Pub. online: 13 Oct 2023 Type: Author Index

Journal: Modern Stochastics: Theory and Applications Volume 10, Issue 4 (2023), p. 459

BDG inequalities and their applications for model-free continuous price paths with instant enforcement

Rafał Marcin Łochowski

https://doi.org/10.15559/23-VMSTA233

Pub. online: 4 Oct 2023 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 10, Issue 4 (2023), pp. 425–457

Abstract

Shafer and Vovk introduce in their book [8] the notion of instant enforcement and instantly blockable properties. However, they do not associate these notions with any outer measure, unlike what Vovk did in the case of sets of “typical” price paths. In this paper an outer measure on the space $[0,+\infty )\times \Omega $ is introduced, which assigns zero value exactly to those sets (properties) of pairs of time t and an elementary event ω which are instantly blockable. Next, for a slightly modified measure, Itô’s isometry and BDG inequalities are proved, and then they are used to define an Itô-type integral. Additionally, few properties are proved for the quadratic variation of model-free continuous martingales, which hold with instant enforcement.

Variance Gamma (nonlocal) equations

Fausto Colantoni

https://doi.org/10.15559/23-VMSTA232

Pub. online: 25 Sep 2023 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 10, Issue 4 (2023), pp. 413–424

Abstract

Some equations are provided for the Variance Gamma process using the definition other than that based on a time-changed Brownian motion. A new nonlocal equation is obtained involving generalized Weyl derivatives, which is true even in the drifted case. The connection to special functions is in focus, and a space equation for the process is studied. In conclusion, the convergence in distribution of a compound Poisson process to the Variance Gamma process is observed.

Critical branching processes in a sparse random environment

Dariusz Buraczewski

Congzao Dong Alexander Iksanov

Alexander Marynych

https://doi.org/10.15559/23-VMSTA231

Pub. online: 29 Aug 2023 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 10, Issue 4 (2023), pp. 397–411

Abstract

We introduce a branching process in a sparse random environment as an intermediate model between a Galton–Watson process and a branching process in a random environment. In the critical case we investigate the survival probability and prove Yaglom-type limit theorems, that is, limit theorems for the size of population conditioned on the survival event.

Perpetual cancellable American options with convertible features

Tsvetelin Zaevski

https://doi.org/10.15559/23-VMSTA230

Pub. online: 4 Aug 2023 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 10, Issue 4 (2023), pp. 367–395

Abstract

The major characteristic of the cancellable American options is the existing writer’s right to cancel the contract prematurely paying some penalty amount. The main purpose of this paper is to introduce and examine a new subclass of such options for which the penalty which the writer owes for this right consists of three parts – a fixed amount, shares of the underlying asset, and a proportion of the usual option payment. We examine the asymptotic case in which the maturity is set to be infinity. We determine the optimal exercise regions for the option’s holder and writer and derive the fair option price.

Multi-mixed fractional Brownian motions and Ornstein–Uhlenbeck processes

Hamidreza Maleki Almani

Tommi Sottinen

https://doi.org/10.15559/23-VMSTA229

Pub. online: 26 Jun 2023 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 10, Issue 4 (2023), pp. 343–366

Abstract

The so-called multi-mixed fractional Brownian motions (mmfBm) and multi-mixed fractional Ornstein–Uhlenbeck (mmfOU) processes are studied. These processes are constructed by mixing by superimposing or mixing (infinitely many) independent fractional Brownian motions (fBm) and fractional Ornstein–Uhlenbeck processes (fOU), respectively. Their existence as ${L^{2}}$ processes is proved, and their path properties, viz. long-range and short-range dependence, Hölder continuity, p-variation, and conditional full support, are studied.

On geometric recurrence for time-inhomogeneous autoregression

Vitaliy Golomoziy

https://doi.org/10.15559/23-VMSTA228

Pub. online: 3 May 2023 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 10, Issue 3 (2023), pp. 313–341

Abstract

The time-inhomogeneous autoregressive model AR(1) is studied, which is the process of the form ${X_{n+1}}={\alpha _{n}}{X_{n}}+{\varepsilon _{n}}$, where ${\alpha _{n}}$ are constants, and ${\varepsilon _{n}}$ are independent random variables. Conditions on ${\alpha _{n}}$ and distributions of ${\varepsilon _{n}}$ are established that guarantee the geometric recurrence of the process. This result is applied to estimate the stability of n-steps transition probabilities for two autoregressive processes ${X^{(1)}}$ and ${X^{(2)}}$ assuming that both ${\alpha _{n}^{(i)}}$, $i\in \{1,2\}$, and distributions of ${\varepsilon _{n}^{(i)}}$, $i\in \{1,2\}$, are close enough.

On some composite Kies families: distributional properties and saturation in Hausdorff sense

Tsvetelin Zaevski

Nikolay Kyurkchiev

https://doi.org/10.15559/23-VMSTA227

Pub. online: 21 Mar 2023 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 10, Issue 3 (2023), pp. 287–312

Abstract

The stochastic literature contains several extensions of the exponential distribution which increase its applicability and flexibility. In the present article, some properties of a new power modified exponential family with an original Kies correction are discussed. This family is defined as a Kies distribution which domain is transformed by another Kies distribution. Its probabilistic properties are investigated and some limitations for the saturation in the Hausdorff sense are derived. Moreover, a formula of a semiclosed form is obtained for this saturation. Also the tail behavior of these distributions is examined considering three different criteria inspired by the financial markets, namely, the VaR, AVaR, and expectile based VaR. Some numerical experiments are provided, too.

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

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