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We introduce a class of continuous Volterra processes, called Volterra clocks, and study their singular limit as the memory kernel collapses to a Dirac mass at zero. The dynamics are parametrised by a function $f$ acting as a nonlinear…

We define and solve Volterra equations driven by an irregular signal, by means of a variant of the rough path theory called algebraic integration. In the Young case, that is for a driving signal with H\"older exponent greater than 1/2, we…

This paper aims at providing statistical guarantees for a kernel based estimation of time varying parameters driving the dynamic of local stationary processes. We extend the results of Dahlhaus et al. (2018) considering the local stationary…

Rochberg's coboundary theorem provides conditions under which the equation $(I-T)y = x$ is solvable in $y$. Here $T$ is a unilateral shift on Hilbert space, $I$ is the identity operator and $x$ is a given vector. The conditions are…

This note aims to give an explicit solution for backward stochastic Volterra integral equations with linear time delayed generators. The process $Y$ is expressed by an integral whose kernel is explicitly given. The processes $Z$ is…

In this paper we consider unbounded solutions of perturbed convolution Volterra summation equations. The equations studied are asymptotically sublinear, in the sense that the state--dependence in the summation is of smaller than linear…

We consider the regularity of sample paths of Volterra-L\'{e}vy processes. These processes are defined as stochastic integrals $$ M(t)=\int_{0}^{t}F(t,r)dX(r), \ \ t \in \mathds{R}_{+}, $$ where $X$ is a L\'{e}vy process and $F$ is a…

We consider the regularity of sample paths of Volterra processes. These processes are defined as stochastic integrals $$ M(t)=\int_{0}^{t}F(t,r)dX(r), \ \ t \in \mathds{R}_{+}, $$ where $X$ is a semimartingale and $F$ is a deterministic…

We consider a class of first-order partial differential operators, acting on the space of ultradifferentiable periodic functions, and we describe their range by using the following conditions on the coefficients of the operators: the…

In this paper, we define an operator function as a series of operators corresponding to the Taylor series representing the function of the complex variable. In previous papers, we considered the case when a function has a decomposition in…

We give upper bounds on the Walsh coefficients of functions for which the derivative of order at least one has bounded variation of fractional order. Further, we also consider the Walsh coefficients of functions in periodic and non-periodic…

Using a new metric for generating rotating solutions, we derive in a general fashion the solution of an imperfect fluid and that of its conformal homolog. We discuss the conditions that the stress-energy tensors and invariant scalars be…

Filter convergence of vector lattice-valued measures is considered, in order to deduce theorems of convergence for their decompositions. First the $\sigma$-additive case is studied, without particular assumptions on the filter; later the…

Many problems of applied mathematics are reduced to the solution of integral equations with special functions in kernels, therefore the inversion formulas for such equations play an important role in solving boundary value problems for…

In this work we prove that a family of explicit numerical finite-difference methods is convergent when applied to a nonlinear Volterra equation with a power-type nonlinearity. In that case the kernel is not of Lipschitz type, therefore the…

We present sufficient conditions for the transience and the existence of local times of a Feller process, and the ultracontractivity of the associated Feller semigroup; these conditions are sharp for L\'{e}vy processes. The proof uses a…

We construct the basis of a stochastic calculus for so-called Volterra processes, i.e., processes which are defined as the stochastic integral of a time-dependent kernel with respect to a standard Brownian motion. For these processes which…

In this work, we study vector-valued functional equations with multiple recursive terms that arise naturally when we are dealing with vector-valued multiplicative Lindley-type recursions. We provide a detailed framework for the solution of…

We propose the numerical methods for solution of the weakly regular linear and nonlinear evolutionary (Volterra) integral equation of the first kind. The kernels of such equations have jump discontinuities along the continuous curves…

Via a coupling argument, it is proved that the solution to a renewal equation has a power law decay rate in the case of a spread out interarrival distribution. By the regenerative property, the convergence in distribution for the recurrence…