Modern Stochastics: Theory and Applications

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.