Modern Stochastics: Theory and Applications

We define power variation estimators for the drift parameter of the stochastic heat equation with the fractional Laplacian and an additive Gaussian noise which is white in time and white or correlated in space. We prove that these estimators are consistent and asymptotically normal and we derive their rate of convergence under the Wasserstein metric.

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.

The problem of (pathwise) large deviations for conditionally continuous Gaussian processes is investigated. The theory of large deviations for Gaussian processes is extended to the wider class of random processes – the conditionally Gaussian processes. The estimates of level crossing probability for such processes are given as an application.

Distance covariance is a quantity to measure the dependence of two random vectors. We show that the original concept introduced and developed by Székely, Rizzo and Bakirov can be embedded into a more general framework based on symmetric Lévy measures and the corresponding real-valued continuous negative definite functions. The Lévy measures replace the weight functions used in the original definition of distance covariance. All essential properties of distance covariance are preserved in this new framework.

We consider a family of mixed processes given as the sum of a fractional Brownian motion with Hurst parameter $H\in (3/4,1)$ and a multiple of an independent standard Brownian motion, the family being indexed by the scaling factor in front of the Brownian motion. We analyze the underlying markets with methods from large financial markets. More precisely, we show the existence of a strong asymptotic arbitrage (defined as in Kabanov and Kramkov [Finance Stoch. 2(2), 143–172 (1998)]) when the scaling factor converges to zero. We apply a result of Kabanov and Kramkov [Finance Stoch. 2(2), 143–172 (1998)] that characterizes the notion of strong asymptotic arbitrage in terms of the entire asymptotic separation of two sequences of probability measures. The main part of the paper consists of proving the entire separation and is based on a dichotomy result for sequences of Gaussian measures and the concept of relative entropy.

A continuous-time regression model with a jointly strictly sub-Gaussian random noise is considered in the paper. Upper exponential bounds for probabilities of large deviations of the least squares estimator for the regression parameter are obtained.

Limit behaviour of temporal and contemporaneous aggregations of independent copies of a stationary multitype Galton–Watson branching process with immigration is studied in the so-called iterated and simultaneous cases, respectively. In both cases, the limit process is a zero mean Brownian motion with the same covariance function under third order moment conditions on the branching and immigration distributions. We specialize our results for generalized integer-valued autoregressive processes and single-type Galton–Watson processes with immigration as well.

Our aim in this paper is to establish some strong stability properties of a solution of a stochastic differential equation driven by a fractional Brownian motion for which the pathwise uniqueness holds. The results are obtained using Skorokhod’s selection theorem.

We investigate large deviation properties of the maximum likelihood drift parameter estimator for Ornstein–Uhlenbeck process driven by mixed fractional Brownian motion.

We show that every multiparameter Gaussian process with integrable variance function admits a Wiener integral representation of Fredholm type with respect to the Brownian sheet. The Fredholm kernel in the representation can be constructed as the unique symmetric square root of the covariance. We analyze the equivalence of multiparameter Gaussian processes by using the Fredholm representation and show how to construct series expansions for multiparameter Gaussian processes by using the Fredholm kernel.