Modern Stochastics: Theory and Applications

In this paper, a non-Gaussian Ornstein–Uhlenbeck process driven by a Hermite–Ornstein–Uhlenbeck process is introduced, which belongs to the qth Wiener chaos. A systematic procedure to identify the drift parameter θ and the Hurst parameter H is given based on the study of the limit behavior of its quadratic variations. Estimators for these two parameters and their asymptotic properties are studied.

An algorithm is proposed for simulation of superpositions of Ornstein–Uhlenbeck processes which may have short- or long-range dependencies and specified marginal distributions. The algorithm is based on the Bondesson–Rosinski representation of the supOU process as a shot-noise process and enables a clear constructive view on the structure of supOU processes. The use of the proposed algorithm is demonstrated for eight positive marginal distributions and eight entire real line marginal distributions when the explicit formulae for the Lévy density are available or not.

In many practical systems, the load changes at the moments when random events occur, which are often modeled as arrivals in a Poisson process independent of the current load state. This modeling approach is widely applicable in areas such as telecommunications, queueing theory, and reliability engineering. This motivates the development of models that combine family-wise scaling with non-Gaussian driving mechanisms, capturing discontinuities or jump-type behavior. In this paper, a stationary time series is formed from increments of a family-wise scaling process defined on the positive real line. This family-wise scaling process is expressed as an integral of a pseudo-Poisson type process. It is established that this stationary time series exhibits long-range dependence, as indicated by an autocovariance function that decays following a power law with a slowly varying component, and a spectral density that displays a power-law divergence at low frequencies. The autocovariances are not summable, indicating strong correlations over long time intervals. This framework extends the classical results on fractional Gaussian noise as well as on series driven by Poisson-type or Lévy-type noise. Additionally, it provides a versatile methodology for the spectral analysis of one-sided long-memory stochastic processes.

The problem of estimating the drift parameter is considered for an Ornstein–Uhlenbeck-type process driven by a tempered fractional Brownian motion (tfBm) or tempered fractional Brownian motion of the second kind (tfBmII). Unlike most existing studies, which assume continuous-time observations, a more realistic setting of discrete-time data is in focus. The strong consistency of a discretized least squares estimator is established under an asymptotic regime where the observation interval tends to zero while the total time horizon increases. A key step in the analysis involves deriving almost sure upper bounds for the increments of both tfBm and tfBmII.

The first-return time is the time that it takes a random walker to go back to the initial position for the first time. In this paper, the first-return time is studied when random walkers perform fractional kinetics, specifically fractional diffusion, that is modelled within the framework of the continuous-time random walk on homogeneous space in the uncoupled formulation with Mittag-Leffler distributed waiting-times. Both the Markovian and non-Markovian settings are considered, as well as any kind of symmetric jump-size distributions, namely with finite or infinite variance. It is shown that the first-return time density is indeed independent of the jump-size distribution when it is symmetric, and therefore it is affected only by the waiting-time distribution that embodies the memory of the process. The analysis is performed in two cases: first jump then wait and first wait then jump, and several exact results are provided, including the relation between results in the Markovian and non-Markovian settings and the difference between the two cases.

Multidimensional generalized backward stochastic differential equations (GBSDEs) are studied within a general filtration that supports a Brownian motion under weak assumptions on the associated data. The existence and uniqueness of solutions in ${\mathbb{L}^{p}}$ for $p\in (1,2)$ are established. The results apply to generators that are stochastic monotone in the y-variable, stochastic Lipschitz in the z-variable, and satisfy a general stochastic linear growth condition.

The cancellable American options, also known as game options, are financial instruments that give a canceling right to the option’s writer in addition to the existing such holder’s right. The writer owes some penalty above the usual option payoff for using this right. We assume that this penalty consists of three parts – a proportion of the usual payoff, some number of shares of the underlying asset, and a fixed amount. It turns out that a cancellable option can be of one of the following three types – a regular American option, an American-style derivative that expires either at the maturity or when the underlying asset reaches the strike, or a real cancellable option. In this paper, the impact of the penalty on the option’s type is investigated. The perpetual case is only explored having in mind that it determines the kind of the finite maturity instruments in some sense.

The area under the receiver operating characteristic curve (AUC) is a suitable measure for the quality of classification algorithms. Here we use the theory of U-statistics in order to derive new confidence intervals for it. The new confidence intervals take into account that only the total sample size used to calculate the AUC can be controlled, while the number of members of the case group and the number of members of the control group are random. We show that the new confidence intervals can not only be used in order to evaluate the quality of the fitted model, but also to judge the quality of the classification algorithm itself. We would like to take this opportunity to show that two popular confidence intervals for the AUC, namely DeLong’s interval and the Mann–Whitney intervals due to Sen, coincide.