Modern Stochastics: Theory and Applications

In this article, we study homogeneous transient diffusion processes. We provide the basic distributions of their local times. It helps to get exact formulas and upper bounds for the moments, exponential moments, and potentials of integral functionals of transient diffusion processes. Some of the results generalize the corresponding results of Salminen and Yor for the Brownian motion with drift.

We provide strong $L_{p}$-rates of approximation of nonsmooth integral-type functionals of Markov processes by integral sums. Our approach is, in a sense, process insensitive and is based on a modification of some well-developed estimates from the theory of continuous additive functionals of Markov processes.

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.

We consider a measurable stationary Gaussian stochastic process. A criterion for testing hypotheses about the covariance function of such a process using estimates for its norm in the space $L_{p}(\mathbb{T})$, $p\ge 1$, is constructed.

Using martingale methods, we provide bounds for the entropy of a probability measure on ${\mathbb{R}}^{d}$ with the right-hand side given in a certain integral form. As a corollary, in the one-dimensional case, we obtain a weighted log-Sobolev inequality.

We deal with a generalization of the classical risk model when an insurance company gets additional funds whenever a claim arrives and consider some practical approaches to the estimation of the ruin probability. In particular, we get an upper exponential bound and construct an analogue to the De Vylder approximation for the ruin probability. We compare results of these approaches with statistical estimates obtained by the Monte Carlo method for selected distributions of claim sizes and additional funds.

We present large sample properties and conditions for asymptotic normality of linear functionals of powers of the periodogram constructed with the use of tapered data.

A mixture with varying concentrations is a modification of a finite mixture model in which the mixing probabilities (concentrations of mixture components) may be different for different observations. In the paper, we assume that the concentrations are known and the distributions of components are completely unknown. Nonparametric technique is proposed for testing hypotheses on functional moments of components.