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In this paper we propose new sufficient conditions for stability of solutions of systems of Volterra linear integral equations and systems of linear integro-differential Volterra equations. Solution stability conditions for systems of…
In this text matrix Volterra integral equation of the first kind is addressed. It is assumed that kernels of the equation have jump discontinuities on non-intersecting curves. Such equations appear in the theory of evolving dynamic systems.…
In this article, we study the ergodicity of neutral retarded stochastic functional differential equations driven by $\alpha$-regular Volterra process. Based on the equivalence between neutral retarded stochastic functional differential…
We establish the existence of martingale solutions to a class of stochastic conservation equations. The underlying models correspond to random perturbations of kinetic models for collective motion such as the Cucker-Smale and Motsch-Tadmor…
Based on a regularized Volterra equation, two different approaches for numerical differentiation are considered. The first approach consists of solving a regularized Volterra equation while the second approach is based on solving a…
We study the Taylor expansion for the solution of a differential equation driven by a multidimensional Holder path with exponent \beta> 1/2. We derive a convergence criterion that enables us to write the solution as an infinite sum of…
We study quadrature methods for solving Volterra integral equations of the first kind with smooth kernels under the presence of noise in the right-hand sides, with the quadrature methods being generated by linear multistep methods. The…
In this paper, we first establish the existence, uniqueness and H\"older continuity of the solution to stochastic Volterra integral equations with weakly singular kernels. Then, we propose a $\theta$-Euler-Maruyama scheme and a Milstein…
We prove existence and uniqueness of a mild solution of a stochastic evolution equation driven by a standard $\alpha$-stable cylindrical L\'evy process defined on a Hilbert space for $\alpha \in (1,2)$. The coefficients are assumed to map…
A stochastic linear transport equation with multiplicative noise is considered and the question of no-blow-up is investigated. The drift is assumed only integrable to a certain power. Opposite to the deterministic case where smooth initial…
We derive unique Banach-valued solutions to stochastic Volterra equations with random coefficients that may depend on pure chance and involve singular kernels. In particular, for controlled and distribution-dependent coefficients these…
We consider stochastic volatility dynamics driven by a general H\"older continuous Volterra-type noise and with unbounded drift. For these so-called SVV-models, we consider the explicit computation of quadratic hedging strategies. While the…
In this article, we construct unique strong solutions to a class of stochastic Volterra differential equations driven by a singular drift vector field and a Wiener noise. Further, we examine the Sobolev differentiability of the strong…
We associate backward and forward Kolmogorov equations to a class of fully nonlinear Stochastic Volterra Equations (SVEs) with convolution kernels $K$ that are singular at the origin. Working on a carefully chosen Hilbert space…
The Duffing oscillator is a paradigm of bistable oscillatory motion in physics, engineering, and biology. Time series of such oscillations are often observed experimentally in a nonlinear system excited by a spontaneously fluctuating force.…
The Volterra integral equations of the first kind with piecewise smooth kernel are considered. Such equations appear in the theory of optimal control of the evolving systems. The existence theorems are proved. The method for constructing…
In the article, some bilinear evolution equations in Hilbert space driven by paths of low regularity are considered and solved explicitly. The driving paths are scalar-valued and continuous, and they are assumed to have a finite $p$-th…
Aim of this work is to extend the results of Cl\'ement, Da Prato & Pr\"uss on the fractional white noise perturbation with Hurst parameter 0<H<1. We will obtain similar results and it will turn out that the regularity of the solution u(t)…
We consider a nonlinear filtering problem for a signal-observation system driven by a Volterra-type Gaussian rough path, whose sample paths may exhibit greater roughness than those of Brownian motion. The observation process includes a…
The stochastic $H^\infty$-norm is defined as the $L^2$-induced norm of the input-output operator of a stochastic linear system. Like the deterministic $H^\infty$-norm it is characterised by a version of the bounded real lemma, but without a…