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A note on randomly stopped sums with zero mean increments

Remigijus Leipus Jonas Šiaulys

https://doi.org/10.15559/23-VMSTA236

Pub. online: 5 Dec 2023 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 11, Issue 1 (2024), pp. 31–42

Abstract

In this paper, the asmptotics is considered for the distribution tail of a randomly stopped sum ${S_{\nu }}={X_{1}}+\cdots +{X_{\nu }}$ of independent identically distributed consistently varying random variables with zero mean, where ν is a counting random variable independent of $\{{X_{1}},{X_{2}},\dots \}$. The conditions are provided for the relation $\mathbb{P}({S_{\nu }}\gt x)\sim \mathbb{E}\nu \hspace{0.1667em}\mathbb{P}({X_{1}}\gt x)$ to hold, as $x\to \infty $, involving the finiteness of $\mathbb{E}|{X_{1}}|$. The result improves that of Olvera-Cravioto [14], where the finiteness of a moment $\mathbb{E}|{X_{1}}{|^{r}}$ for some $r\gt 1$ was assumed.

Parameter estimation for fractional mixed fractional Brownian motion based on discrete observations

Kostiantyn Ralchenko

Mykyta Yakovliev

https://doi.org/10.15559/23-VMSTA234

Pub. online: 5 Dec 2023 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 11, Issue 1 (2024), pp. 1–29

Abstract

The object of investigation is the mixed fractional Brownian motion of the form ${X_{t}}=\kappa {B_{t}^{{H_{1}}}}+\sigma {B_{t}^{{H_{2}}}}$, driven by two independent fractional Brownian motions ${B_{1}^{H}}$ and ${B_{2}^{H}}$ with Hurst parameters ${H_{1}}\lt {H_{2}}$. Strongly consistent estimators of unknown model parameters ${({H_{1}},{H_{2}},{\kappa ^{2}},{\sigma ^{2}})^{\top }}$ are constructed based on the equidistant observations of a trajectory. Joint asymptotic normality of these estimators is proved for $0\lt {H_{1}}\lt {H_{2}}\lt \frac{3}{4}$.

A quantitative functional central limit theorem for shallow neural networks

Valentina Cammarota Domenico Marinucci Michele Salvi

Stefano Vigogna

https://doi.org/10.15559/23-VMSTA238

Pub. online: 28 Nov 2023 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 11, Issue 1 (2024), pp. 85–108

Abstract

We prove a quantitative functional central limit theorem for one-hidden-layer neural networks with generic activation function. Our rates of convergence depend heavily on the smoothness of the activation function, and they range from logarithmic for nondifferentiable nonlinearities such as the ReLu to $\sqrt{n}$ for highly regular activations. Our main tools are based on functional versions of the Stein–Malliavin method; in particular, we rely on a quantitative functional central limit theorem which has been recently established by Bourguin and Campese [Electron. J. Probab. 25 (2020), 150].

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.

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

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