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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.

Consistency of LSE for the many-dimensional symmetric textured surface parameters

Oleksandr Dykyi

Alexander Ivanov

https://doi.org/10.15559/23-VMSTA225

Pub. online: 16 Mar 2023 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 10, Issue 3 (2023), pp. 267–285

Abstract

A multivariate trigonometric regression model is considered. In the paper strong consistency of the least squares estimator for amplitudes and angular frequencies is obtained for such a multivariate model on the assumption that the random noise is a homogeneous or homogeneous and isotropic Gaussian, specifically, strongly dependent random field on ${\mathbb{R}^{M}},M\ge 3$.

Ruin probabilities as functions of the roots of a polynomial

David J. Santana

Luis Rincón

https://doi.org/10.15559/23-VMSTA226

Pub. online: 15 Mar 2023 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 10, Issue 3 (2023), pp. 247–266

Abstract

A new formula for the ultimate ruin probability in the Cramér–Lundberg risk process is provided when the claims are assumed to follow a finite mixture of m Erlang distributions. Using the theory of recurrence sequences, the method proposed here shifts the problem of finding the ruin probability to the study of an associated characteristic polynomial and its roots. The found formula is given by a finite sum of terms, one for each root of the polynomial, and allows for yet another approximation of the ruin probability. No constraints are assumed on the multiplicity of the roots and that is illustrated via a couple of numerical examples.

The Burgers equation driven by a stochastic measure

Vadym Radchenko

https://doi.org/10.15559/23-VMSTA224

Pub. online: 21 Feb 2023 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 10, Issue 3 (2023), pp. 229–246

Abstract

The class of one-dimensional equations driven by a stochastic measure μ is studied. For μ only σ-additivity in probability is assumed. This class of equations includes the Burgers equation and the heat equation. The existence and uniqueness of the solution are proved, and the averaging principle for the equation is studied.

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

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