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Strong laws of large numbers for lightly trimmed sums of generalized Oppenheim expansions

Rita Giuliano

Milto Hadjikyriakou

https://doi.org/10.15559/25-VMSTA272

Pub. online: 13 Feb 2025 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications

Abstract

In the framework of generalized Oppenheim expansions, almost sure convergence results for lightly trimmed sums are proven. First, a particular class of expansions is identified for which a convergence result is proven assuming that only the largest summand is deleted from the sum; this result generalizes a strong law recently proven for the Lüroth digits and also covers some new cases that have never been studied before. Next, any assumptions concerning the structure of the Oppenheim expansions are dropped and a result concerning trimmed sums is proven when at least two summands are trimmed; combining this latter theorem with the asymptotic behavior of the r-th maximum term of the expansion, a convergence result is obtained for the case in which only the largest summand is deleted from the sum.

About stability of equilibria of one system of stochastic delay differential equations with exponential nonlinearity

Leonid Shaikhet

https://doi.org/10.15559/25-VMSTA274

Pub. online: 13 Feb 2025 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications

Abstract

A system of two nonlinear delay differential equations under stochastic perturbations is considered. Nonlinearity of the exponential type in each equation of the system under consideration depends on the both variables of the system. The stability in probability of the zero and nonzero equilibria of the system is studied via the general method of Lyapunov functionals construction and the method of linear matrix inequalities (LMIs). The obtained results are illustrated via examples and figures with numerical simulations of solutions of a considered system of stochastic differential equations. The proposed way of investigation can be applied to nonlinear systems of higher dimension and with other types of nonlinearity, both for delay differential equations and for difference equations.

Skorokhod

M_{1}

convergence of maxima of multivariate linear processes with heavy-tailed innovations and random coefficients

Danijel Krizmanić

https://doi.org/10.15559/25-VMSTA271

Pub. online: 28 Jan 2025 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications

Abstract

In this paper, functional convergence is derived for the partial maxima stochastic processes of multivariate linear processes with weakly dependent heavy-tailed innovations and random coefficients. The convergence takes place in the space of ${\mathbb{R}^{d}}$-valued càdlàg functions on $[0,1]$ endowed with the weak Skorokhod ${M_{1}}$ topology.

Exponential utility maximization in small/large financial markets

Miklós Rásonyi

Hasanjan Sayit

https://doi.org/10.15559/24-VMSTA270

Pub. online: 23 Jan 2025 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications

Abstract

Obtaining a utility-maximizing optimal portfolio in a closed form is a challenging issue when the return vector follows a more general distribution than the normal one. In this paper, for markets based on finitely many assets, a closed-form expression is given for optimal portfolios that maximize an exponential utility function when the return vector follows normal mean-variance mixture models. Especially, the used approach expresses the closed-form solution in terms of the Laplace transformation of the mixing distribution of the normal mean-variance mixture model and no distributional assumptions on the mixing distribution are made.

Also considered are large financial markets based on normal mean-variance mixture models, and it is shown that the optimal exponential utilities in small markets converge to the optimal exponential utility in the large financial market. This shows, in particular, that to reach the best utility level investors need to diversify their investments to include infinitely many assets into their portfolio, and with portfolios based on only finitely many assets they will never be able to reach the optimum level of utility.

Noncentral moderate deviations for time-changed Lévy processes with inverse of stable subordinators

Antonella Iuliano

Claudio Macci

Alessandra Meoli

https://doi.org/10.15559/24-VMSTA269

Pub. online: 31 Dec 2024 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications

Abstract

This paper presents some extensions of recent noncentral moderate deviation results. In the first part, the results in [Statist. Probab. Lett. 185, Paper No. 109424, 8 pp. (2022)] are generalized by considering a general Lévy process $\{S(t):t\ge 0\}$ instead of a compound Poisson process. In the second part, it is assumed that $\{S(t):t\ge 0\}$ has bounded variation and is not a subordinator; thus $\{S(t):t\ge 0\}$ can be seen as the difference of two independent nonnull subordinators. In this way, the results in [Mod. Stoch. Theory Appl. 11, 43–61] for Skellam processes are generalized.

Regularity of paths of stochastic measures

Vadym Radchenko

https://doi.org/10.15559/24-VMSTA268

Pub. online: 26 Nov 2024 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications

Abstract

Random functions $\mu (x)$, generated by values of stochastic measures are considered. The Besov regularity of the continuous paths of $\mu (x)$, $x\in {[0,1]^{d}}$, is proved. Fourier series expansion of $\mu (x)$, $x\in [0,2\pi ]$, is obtained. These results are proved under weaker conditions than similar results in previous papers.

Statistical inference for nth-order mixed fractional Brownian motion with polynomial drift

Mohamed El Omari

https://doi.org/10.15559/24-VMSTA267

Pub. online: 19 Nov 2024 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications

Abstract

The mixed model with polynomial drift of the form $X(t)=\theta \mathcal{P}(t)+\alpha W(t)+\sigma {B_{H}^{n}}(t)$ is studied, where ${B_{H}^{n}}$ is the nth-order fractional Brownian motion with Hurst index $H\in (n-1,n)$ and $n\ge 2$, independent of the Wiener process W. The polynomial function $\mathcal{P}$ is known, with degree $d(\mathcal{P})\in [1,n)$. Based on discrete observations and using the ergodic theorem estimates of H, ${\alpha ^{2}}$ and ${\sigma ^{2}}$ are given. Finally, a continuous time maximum likelihood estimator of θ is provided. Both strong consistency and asymptotic normality of the proposed estimators are established.

Subdiffusive option price model with Inverse Gaussian subordinator

Nataliya Shchestyuk

Sergii Tyshchenko

https://doi.org/10.15559/24-VMSTA265

Pub. online: 12 Nov 2024 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications

Abstract

The paper focuses on the option price subdiffusive model under the unusual behavior of the market, when the price may not be changed for some time, which is a quite common situation in modern illiquid financial markets or during global crises. In the model, the risk-free bond motion and classical geometrical Brownian motion (GBM) are time-changed by an inverted inverse Gaussian($\mathit{IG}$) subordinator. We explore the correlation structure of the subdiffusive GBM stock returns process, discuss option pricing techniques based on the martingale option pricing method and the fractal Dupire equation, and demonstrate how it applies in the case of the $\mathit{IG}$ subordinator.

Bivariate dependence, stochastic orders and conditional tails of the recurrence times in a renewal process

Sotirios Losidis Konstadinos Politis

Georgios Psarrakos

https://doi.org/10.15559/24-VMSTA266

Pub. online: 12 Nov 2024 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications

Abstract

The structure of dependence between the forward and the backward recurrence times in a renewal process is considered. Monotonicity properties, as a function of time, for the tail of the bivariate distribution for the recurrence times are discussed, as well as their link with aging properties of the interarrival distribution F. A necessary and sufficient condition for the renewal function to be concave is also obtained. Finally, some properties of the conditional tail for one of the two recurrence times, given some information on the other, are studied. The results are illustrated by some numerical examples.

2020 Mathematics Subject Classification index

https://doi.org/10.15559/24-VMSTA114MI

Pub. online: 21 Aug 2024 Type: 2020 Mathematics Subject Classification Index

Journal: Modern Stochastics: Theory and Applications Volume 11, Issue 4 (2024), pp. 495–498

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