We present a new construction of the normal inverse Gaussian (NIG) fractal activity time model for a risky asset. The construction uses superpositions of diffusion processes and allows for specified exact NIG marginal distributions of the returns and flexible and tractable dependence structure...

This paper presents the log-normal, log-gamma and log-negative inverted gamma scenarios in limit theorems for multifractal products of stochastic processes with long-range dependence.

This paper presents a class of minimum contrast estimators for stochastic processes with possible long-range dependence based on the information on higher-order spectral densities. The results on consistency and asymptotic normality of the proposed estimators are provided.

Continuous version of the Whittle contrast functional supplied with a specific weight function is proposed for the estimation of continuous-parameter stochastic processes and fields observed continuously. The results on consistency and asymptotic normality of the estimators are provided.

Gaussian and non-Gaussian limiting distributions of the rescaled solutions of the fractional (in time) diffusion-wave equation for Gaussian and non-Gaussian initial data with long-range dependence are described in terms of multiple Wiener-Itô integrals.

Non-Gaussian limiting distributions of the rescaling solutions of the heat equation for non-Gaussian initial data with long-range dependence are discribed in terms of their multiple stochastic integral representations.

The rate of convergence (in the uniform Kolmogorov's distance) for probability distributions of parabolically rescaled solutions of the multidimensional Burgers' equation with random singular Gaussian initial data (with long-range dependence) to a limit Gaussian random field is discussed in this...