Modern Stochastics: Theory and Applications

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.

The stochastic transport equation is considered where the randomness is given by a symmetric integral with respect to a stochastic measure. For a stochastic measure, only σ-additivity in probability and continuity of paths is assumed. Existence and uniqueness of a weak solution to the equation are proved.

The paper is devoted to a stochastic heat equation with a mixed fractional Brownian noise. We investigate the covariance structure, stationarity, upper bounds and asymptotic behavior of the solution. Based on its discrete-time observations, we construct a strongly consistent estimator for the Hurst index H and prove the asymptotic normality for $H<3>. Then assuming the parameter H to be known, we deal with joint estimation of the coefficients at the Wiener process and at the fractional Brownian motion. The quality of estimators is illustrated by simulation experiments.

The paper presents the study on the existence and uniqueness (strong and in law) of a class of non-Markovian SDEs whose drift contains the derivative in the sense of distributions of a continuous function.

This paper deals with linear stochastic partial differential equations with variable coefficients driven by Lévy white noise. First, an existence theorem for integral transforms of Lévy white noise is derived and the existence of generalized and mild solutions of second order elliptic partial differential equations is proved. Further, the generalized electric Schrödinger operator for different potential functions V is discussed.

We consider the cable equation in the mild form driven by a general stochastic measure. The averaging principle for the equation is established. The rate of convergence is estimated. The regularity of the mild solution is also studied. The orders in time and space variables in the Holder condition for the solution are improved in comparison with previous results in the literature on this topic.

The paper deals with a stochastic heat equation driven by an additive fractional Brownian space-only noise. We prove that a solution to this equation is a stationary and ergodic Gaussian process. These results enable us to construct a strongly consistent estimator of the diffusion parameter.

We introduce a stochastic partial differential equation (SPDE) with elliptic operator in divergence form, with measurable and bounded coefficients and driven by space-time white noise. Such SPDEs could be used in mathematical modelling of diffusion phenomena in medium consisting of different kinds of materials and undergoing stochastic perturbations. We characterize the solution and, using the Stein–Malliavin calculus, we prove that the sequence of its recentered and renormalized spatial quadratic variations satisfies an almost sure central limit theorem. Particular focus is given to the interesting case where the coefficients of the operator are piecewise constant.

A moderate deviations principle for the law of a stochastic Burgers equation is proved via the weak convergence approach. In addition, some useful estimates toward a central limit theorem are established.

For a class of non-autonomous parabolic stochastic partial differential equations defined on a bounded open subset $D\subset {\mathbb{R}^{d}}$ and driven by an ${L^{2}}(D)$-valued fractional Brownian motion with the Hurst index $H>1/2$, a new result on existence and uniqueness of a mild solution is established. Compared to the existing results, the uniqueness in a fully nonlinear case is shown, not assuming the coefficient in front of the noise to be affine. Additionally, the existence of moments for the solution is established.