Modern Stochastics: Theory and Applications

A Markov process defined by some pseudo-differential operator of an order $1\lt \alpha \lt 2$ as the process generator is considered. Using a pseudo-gradient operator, that is, the operator defined by the symbol $i\lambda |\lambda {|^{\beta -1}}$ with some $0\lt \beta \lt 1$, the perturbation of the Markov process under consideration by the pseudo-gradient with a multiplier, which is integrable at some large enough power, is constructed. Such perturbation defines a family of evolution operators, properties of which are investigated. A corresponding Cauchy problem is considered.

Despite the relevance of the binomial distribution for probability theory and applied statistical inference, its higher-order moments are poorly understood. The existing formulas are either not general enough, or not structured and simplified enough for intended applications.

The test for the location of the tangency portfolio on the set of feasible portfolios is proposed when both the population and the sample covariance matrices of asset returns are singular. The particular case of investigation is when the number of observations, n, is smaller than the number of assets, k, in the portfolio, and the asset returns are i.i.d. normally distributed with singular covariance matrix Σ such that $rank(\boldsymbol{\Sigma })=r\lt n\lt k+1$. The exact distribution of the test statistic is derived under both the null and alternative hypotheses. Furthermore, the high-dimensional asymptotic distribution of that test statistic is established when both the rank of the population covariance matrix and the sample size increase to infinity so that $r/n\to c\in (0,1)$. Theoretical findings are completed by comparing the high-dimensional asymptotic test with an exact finite sample test in the numerical study. A good performance of the obtained results is documented. To get a better understanding of the developed theory, an empirical study with data on the returns on the stocks included in the S&P 500 index is provided.

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.

This paper establishes the conditions for the existence of a stationary solution to the first-order autoregressive equation on a plane as well as properties of the stationary solution. The first-order autoregressive model on a plane is defined by the equation A stationary solution X to the equation exists if and only if $(1-a-b-c)(1-a+b+c)(1+a-b+c)(1+a+b-c)\gt 0$. The stationary solution X satisfies the causality condition with respect to the white noise ϵ if and only if $1-a-b-c\gt 0$, $1-a+b+c\gt 0$, $1+a-b+c\gt 0$ and $1+a+b-c\gt 0$. A sufficient condition for X to be purely nondeterministic is provided.