Modern Stochastics: Theory and Applications

The time-inhomogeneous autoregressive model AR(1) is studied, which is the process of the form ${\mathit{X}}_{\mathit{n}+1}={\mathit{\alpha }}_{\mathit{n}}{\mathit{X}}_{\mathit{n}}+{\mathit{\epsilon }}_{\mathit{n}}$, where ${\mathit{\alpha }}_{\mathit{n}}$ are constants, and ${\mathit{\epsilon }}_{\mathit{n}}$ are independent random variables. Conditions on ${\mathit{\alpha }}_{\mathit{n}}$ and distributions of ${\mathit{\epsilon }}_{\mathit{n}}$ are established that guarantee the geometric recurrence of the process. This result is applied to estimate the stability of n-steps transition probabilities for two autoregressive processes ${\mathit{X}}^{\mathrm{(}1\mathrm{)}}$ and ${\mathit{X}}^{\mathrm{(}2\mathrm{)}}$ assuming that both ${\mathit{\alpha }}_{\mathit{n}}^{\mathrm{(}\mathit{i}\mathrm{)}}$, $\mathit{i}\in \{1\mathrm{,}2\}$, and distributions of ${\mathit{\epsilon }}_{\mathit{n}}^{\mathrm{(}\mathit{i}\mathrm{)}}$, $\mathit{i}\in \{1\mathrm{,}2\}$, are close enough.

The stochastic literature contains several extensions of the exponential distribution which increase its applicability and flexibility. In the present article, some properties of a new power modified exponential family with an original Kies correction are discussed. This family is defined as a Kies distribution which domain is transformed by another Kies distribution. Its probabilistic properties are investigated and some limitations for the saturation in the Hausdorff sense are derived. Moreover, a formula of a semiclosed form is obtained for this saturation. Also the tail behavior of these distributions is examined considering three different criteria inspired by the financial markets, namely, the VaR, AVaR, and expectile based VaR. Some numerical experiments are provided, too.

A multivariate trigonometric regression model is considered. In the paper strong consistency of the least squares estimator for amplitudes and angular frequencies is obtained for such a multivariate model on the assumption that the random noise is a homogeneous or homogeneous and isotropic Gaussian, specifically, strongly dependent random field on ${\mathbb{R}^{M}},M\ge 3$.

A new formula for the ultimate ruin probability in the Cramér–Lundberg risk process is provided when the claims are assumed to follow a finite mixture of m Erlang distributions. Using the theory of recurrence sequences, the method proposed here shifts the problem of finding the ruin probability to the study of an associated characteristic polynomial and its roots. The found formula is given by a finite sum of terms, one for each root of the polynomial, and allows for yet another approximation of the ruin probability. No constraints are assumed on the multiplicity of the roots and that is illustrated via a couple of numerical examples.

The class of one-dimensional equations driven by a stochastic measure μ is studied. For μ only σ-additivity in probability is assumed. This class of equations includes the Burgers equation and the heat equation. The existence and uniqueness of the solution are proved, and the averaging principle for the equation is studied.