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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 Volume 12, Issue 2 (2025), pp. 169–187

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.

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 Volume 12, Issue 2 (2025), pp. 153–167

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.

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 Volume 12, Issue 2 (2025), pp. 135–152

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.

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

Keywords index

https://doi.org/10.15559/24-VMSTA114KI

Pub. online: 21 Aug 2024 Type: Keywords Index

Journal: Modern Stochastics: Theory and Applications Volume 11, Issue 4 (2024), pp. 493–494

Author index

https://doi.org/10.15559/24-VMSTA114AI

Pub. online: 21 Aug 2024 Type: Author Index

Journal: Modern Stochastics: Theory and Applications Volume 11, Issue 4 (2024), pp. 491–492

A note on optimal liquidation with linear price impact

Yan Dolinsky

Doron Greenstein

https://doi.org/10.15559/24-VMSTA264

Pub. online: 20 Aug 2024 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 12, Issue 2 (2025), pp. 123–134

Abstract

In this note the maximization of the expected terminal wealth for the setup of quadratic transaction costs is considered. First, a very simple probabilistic solution to the problem is provided. Although the problem was largely studied, as far as authors know up to date this simple and probabilistic form of the solution has not appeared in the literature. Next, the general result is applied for the numerical study of the case where the risky asset is given by a fractional Brownian motion and the information flow of the investor can be diversified.

First-order planar autoregressive model

Sergiy Shklyar

https://doi.org/10.15559/24-VMSTA263

Pub. online: 6 Aug 2024 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 12, Issue 1 (2025), pp. 83–121

Abstract

This paper establishes the conditions for the existence of a stationary solution to the first-order autoregressive equation on a plane as well as properties of the stationary solution. The first-order autoregressive model on a plane is defined by the equation

\[ {X_{i,j}}=a{X_{i-1,j}}+b{X_{i,j-1}}+c{X_{i-1,j-1}}+{\epsilon _{i,j}}.\]

A stationary solution X to the equation exists if and only if $(1-a-b-c)(1-a+b+c)(1+a-b+c)(1+a+b-c)\gt 0$. The stationary solution X satisfies the causality condition with respect to the white noise ϵ if and only if $1-a-b-c\gt 0$, $1-a+b+c\gt 0$, $1+a-b+c\gt 0$ and $1+a+b-c\gt 0$. A sufficient condition for X to be purely nondeterministic is provided.

An explicit expression for the autocovariance function of X on the axes is provided. With Yule–Walker equations, this facilitates the computation of the autocovariance function everywhere, at all integer points of the plane. In addition, all situations are described where different parameters determine the same autocovariance function of X.

Handy formulas for binomial moments

Maciej Skorski

https://doi.org/10.15559/24-VMSTA260

Pub. online: 30 Jul 2024 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 12, Issue 1 (2025), pp. 27–41

Abstract

Despite the relevance of the binomial distribution for probability theory and applied statistical inference, its higher-order moments are poorly understood. The existing formulas are either not general enough, or not structured and simplified enough for intended applications.

This paper introduces novel formulas for binomial moments in the form of polynomials in the variance rather than in the success probability. The obtained formulas are arguably better structured, simpler and superior in their numerical properties compared to prior works. In addition, the paper presents algorithms to derive these formulas along with working implementation in Python’s symbolic algebra package.

The novel approach is a combinatorial argument coupled with clever algebraic simplifications which rely on symmetrization theory. As an interesting byproduct asymptotically sharp estimates for central binomial moments are established, improving upon previously known partial results.

Heat equation with a general stochastic measure in a bounded domain

Boris Manikin

https://doi.org/10.15559/24-VMSTA262

Pub. online: 12 Jul 2024 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 12, Issue 1 (2025), pp. 61–82

Abstract

A stochastic heat equation on $[0,T]\times B$, where B is a bounded domain, is considered. The equation is driven by a general stochastic measure, for which only σ-additivity in probability is assumed. The existence, uniqueness and Hölder regularity of the solution are proved.

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