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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 Volume 12, Issue 2 (2025), pp. 225–250

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 Volume 12, Issue 2 (2025), pp. 203–224

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 Volume 12, Issue 2 (2025), pp. 189–201

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 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.

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