Abstract In this paper, we introduce the notion of quasi- ( m , n ) (m,n) -paranormal operators on a Hilbert space and prove basic structural properties for the same class of operators. We also characterize these operators. We prove that if 𝑇 is quasi- ( m , n ) (m,n) -paranormal, then the spectral mapping theorem holds, that is, f ⁢ ( w ⁢ ( T ) ) = w ⁢ ( f ⁢ ( T ) ) f(w(T))=w(f(T)) for every analytic function f ∈ H ⁢ ( σ ⁢ ( T ) ) f\in\mathcal{H}(\sigma(T)) . We also show more general results for operators in the class.

Abstract This article explores the averaging principle for 𝜓-Hilfer fractional neutral impulsive stochastic differential equations with pantograph-type delay driven by Poisson jumps. With the help of the Khasḿinskii approach, we approximate the nonautonomous 𝜓-Hilfer fractional stochastic differential equations with both deterministic and stochastic jumps by an autonomous 𝜓-Hilfer fractional stochastic differential equations without deterministic jumps and to demonstrate the convergence in L p L^{\mathtt{p}} sense. Using Lipschitz and growth conditions and elementary inequalities, assumptions are made. With the help of Jensen’s inequality, Burkholder–Davis–Gundy inequality, Hölder inequality, Doob’s martingale inequality, Kunitha’s inequality and Gronwall–Bellman’s inequality, an averaging principle of our proposed system is obtained in the sense of L p \mathtt{L^{p}} convergence. An illustration is given to support the theoretical results.

Abstract Let Φ ′ \Phi^{\prime} denote the strong dual of a nuclear space Φ and let C ∞ ⁢ ( Φ ′ ) C_{\infty}(\Phi^{\prime}) be the collection of all continuous mappings x : [ 0 , ∞ ) → Φ ′ x\colon[0,\infty)\rightarrow\Phi^{\prime} equipped with the topology of local uniform convergence. In this paper, we prove sufficient conditions for tightness of probability measures on C ∞ ⁢ ( Φ ′ ) C_{\infty}(\Phi^{\prime}) and for weak convergence in C ∞ ⁢ ( Φ ′ ) C_{\infty}(\Phi^{\prime}) for a sequence of Φ ′ \Phi^{\prime} -valued processes. We illustrate our results with two applications. First, we show the central limit theorem for local martingales taking values in the dual of an ultrabornological nuclear space. Second, we prove sufficient conditions for the weak convergence in C ∞ ⁢ ( Φ ′ ) C_{\infty}(\Phi^{\prime}) for a sequence of solutions to stochastic partial differential equations driven by semimartingale noise.

Abstract We prove the global elliptic law for random non-Hermitian matrices with complex covariances of its entries. The proof of the global elliptic law is based on generalized Lindeberg condition, VICTORIA transform, RESPECT method and new limit theorems.

Abstract In this study, we derive the necessary conditions for achieving controllability in a particular group of neutral stochastic integro-differential evolution equations. These equations are driven by a fractional Brownian motion with a Hurst parameter that is less than 1/2. To obtain these results, we utilize semigroup theory, resolvent operators, and a fixed-point technique. Furthermore, we provide an application of these findings to neutral partial integro-differential stochastic equations that have been perturbed by fractional Brownian motion.

Abstract We study the problem of misspecification when the stochastic model proposed by the statistician (theoretical model), through a stochastic differential equation, has a change point type singularity in the drift but the real model has a smooth drift function and the driving force is a fractional Brownian motion.

Abstract In this paper, we investigate the existence and uniqueness of the solution for a system of nonlinear, fully coupled forward-backward doubly stochastic differential equations with Poisson jumps. Our study is conducted within the framework of a separable real Hilbert space, and we employ a continuation method to establish the proof.

Abstract After leaving fixed the environment of the Brox diffusion, we give explicitly the probability density of the local time of this process at first passage times. The main idea is to use the fact that the Brox diffusion can be written in term of a time change of a standard Brownian motion. Working with a specific stopping times is key.

Abstract Motivated by the truncated Euler–Maruyama method developed in [X. Mao, Convergence rates of the truncated Euler–Maruyama method for stochastic differential equations, J. Comput. Appl. Math. 296 2016, 362–375], we will study the truncated Euler–Maruyama method of a class of perturbed stochastic differential equations with reflected boundary (in short PSDERBs). The main objective of this paper is to established the rate of strong convergence under the drift coefficient satisfies a one-sided Lipschitz condition plus the Khasminskii-type condition.