Modern Stochastics: Theory and Applications

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.

The integral with respect to a multidimensional stochastic measure, assuming only its σ-additivity in probability, is studied. The continuity and differentiability of realizations of the integral are established.

In this article, a non-Gaussian long memory process is constructed by the aggregation of independent copies of a fractional Lévy Ornstein–Uhlenbeck process with random coefficients. Several properties and a limit theorem are studied for this new process. Finally, some simulations of the limit process are shown.

This paper provides a multivariate extension of Bertoin’s pathwise construction of a Lévy process conditioned to stay positive or negative. Thus obtained processes conditioned to stay in half-spaces are closely related to the original process on a compact time interval seen from its directional extremal points. In the case of a correlated Brownian motion the law of the conditioned process is obtained by a linear transformation of a standard Brownian motion and an independent Bessel-3 process. Further motivation is provided by a limit theorem corresponding to zooming in on a Lévy process with a Brownian part at the point of its directional infimum. Applications to zooming in at the point furthest from the origin are envisaged.

In this paper the study of a three-parametric class of Gaussian Volterra processes is continued. This study was started in Part I of the present paper. The class under consideration is a generalization of a fractional Brownian motion that is in fact a one-parametric process depending on Hurst index H. On the one hand, the presence of three parameters gives us a freedom to operate with the processes and we get a wider application possibilities. On the other hand, it leads to the need to apply rather subtle methods, depending on the intervals where the parameters fall. Integration with respect to the processes under consideration is defined, and it is found for which parameters the processes are differentiable. Finally, the Volterra representation is inverted, that is, the representation of the underlying Wiener process via Gaussian Volterra process is found. Therefore, it is shown that for any indices for which Gaussian Volterra process is defined, it generates the same flow of sigma-fields as the underlying Wiener process – the property that has been used many times when considering a fractional Brownian motion.

The stochastic process of the form is considered, where W is a standard Wiener process, $\alpha >-\frac{1}{2}$, $\gamma >-1$, and $\alpha +\beta +\gamma >-\frac{3}{2}$. It is proved that the process X is well-defined and continuous. The asymptotic properties of the variances and bounds for the variances of the increments of the process X are studied. It is also proved that the process X satisfies the single-point Hölder condition up to order $\alpha +\beta +\gamma +\frac{3}{2}$ at point 0, the “interval” Hölder condition up to order $\min \big(\gamma +\frac{3}{2},\hspace{0.2222em}1\big)$ on the interval $[{t_{0}},T]$ (where $0<{t_{0}}<T$), and the Hölder condition up to order $\min \big(\alpha +\beta +\gamma +\frac{3}{2},\hspace{0.2778em}\gamma +\frac{3}{2},\hspace{0.2778em}1\big)$ on the entire interval $[0,T]$.

This note provides a simple sufficient condition ensuring that solutions of stochastic delay differential equations (SDDEs) driven by subordinators are nonnegative. While, to the best of our knowledge, no simple nonnegativity conditions are available in the context of SDDEs, we compare our result to the literature within the subclass of invertible continuous-time ARMA (CARMA) processes. In particular, we analyze why our condition cannot be necessary for CARMA($p,q$) processes when $p=2$, and we show that there are various situations where our condition applies while existing results do not as soon as $p\ge 3$. Finally, we extend the result to a multidimensional setting.

A stochastic heat equation on $[0,T]\times \mathbb{R}$ driven by a general stochastic measure $d\mu (t)$ is investigated in this paper. For the integrator μ, we assume the σ-additivity in probability only. The existence, uniqueness, and Hölder regularity of the solution are proved.

Our paper starts from presentation and comparison of three definitions for the self-similar field. The interconnection between these definitions has been established. Then we consider the Lamperti scaling transformation for the self-similar field and investigate the connection between the scaling transformation for such field and the shift transformation for the corresponding stationary field. It was also shown that the fractional Brownian sheet has the ergodic scaling transformation. The strong limit theorems for the anisotropic growth of the sample paths of the self-similar field at 0 and at ∞ for the upper and lower functions have been proved. It was obtained the upper bound for growth of the field with ergodic scaling transformation for slowly varying functions. We present some examples of iterated log-type limits for the Gaussian self-similar random fields.