Volterra difference equations

This dissertation consists of five papers in which discrete Volterra equations of different types and orders are studied and results regarding the behavior of their solutions are established. The first paper presents some fundamental results about subexponential sequences. It also illustrates the subexponential solutions of scalar linear Volterra sum-difference equations are asymptotically stable. The exact value of the rate of convergence of asymptotically stable solutions is found by determining the asymptotic behavior of the transient renewal equations. The study of subexponential solutions is also continued in the second and third articles. The second paper investigates the same equation using the same process as considered in the first paper. The discussion focuses on a positive lower bound of the rate of convergence of the asymptotically stable solutions. The third paper addresses the rate of convergence of the solutions of scalar linear Volterra sum-difference equations with delay. The result is proved by developing the rate of convergence of transient renewal delay difference equations. The fourth paper discusses the existence of bounded solutions on an unbounded domain of more general nonlinear Volterra sum-difference equations using the Schaefer fixed point theorem and the Lyapunov direct method. The fifth paper examines the asymptotic behavior of nonoscillatory solutions of higher-order integro-dynamic equations and establishes some new criteria based on so-called time scales, which unifies and extends both discrete and continuous mathematical analysis. Beside these five research papers that focus on discrete Volterra equations, this dissertation also contains an introduction, a section on difference calculus, a section on time scales calculus, and a conclusion. --Abstract, page v

Some stability conditions for scalar Volterra difference equations

New explicit stability results are obtained for the following scalar linear difference equation $$x(n+1)-x(n)=-a(n)x(n)+\sum_{k=1}^n A(n,k)x(k)+f(n)$$ and for some nonlinear Volterra difference equations

Existence of resolvent for conformable fractional Volterra integral equations

In this paper, we consider the conformable fractional Volterra integral equation. We study the existence of a resolvent kernel corresponding to conformable fractional Volterra integral equation. The technique of proof involves Lebesgue dominated convergence theorem. Our results improve and extend the results obtained in literature

Qualitative approximation of solutions to discrete Volterra equations

We present a new approach to the theory of asymptotic properties of solutions to discrete Volterra equations of the form \begin{equation*} \Delta^m x_n=b_n+\sum_{k=1}^{n}K(n,k)f(k,x_{\sigma(k)}). \end{equation*} Our method is based on using the iterated remainder operator and asymptotic difference pairs. This approach allows us to control the degree of approximation

Stochastic delay difference and differential equations: applications to financial markets

This thesis deals with the asymptotic behaviour of stochastic difference and functional differential equations of Itˆo type. Numerical methods which both minimise error and preserve asymptotic features of the underlying continuous equation are studied. The equations have a form which makes them suitable to model financial markets in which agents use past prices. The second chapter deals with the behaviour of moving average models of price formation. We show that the asset returns are positively and exponentially correlated, while the presence of feedback traders causes either excess volatility or a market bubble or crash. These results are robust to the presence of nonlinearities in the traders’ demand functions. In Chapters 3 and 4, we show that these phenomena persist if trading takes place continuously by modelling the returns using linear and nonlinear stochastic functional differential equations (SFDEs). In the fifth chapter, we assume that some traders base their demand on the difference between current returns and the maximum return over several trading periods, leading to an analysis of stochastic difference equations with maximum functionals. Once again it is shown that prices either fluctuate or undergo a bubble or crash. In common with the earlier chapters, the size of the largest fluctuations and the growth rate of the bubble or crash is determined. The last three chapters are devoted to the discretisation of the SFDE presented in Chapter 4. Chapter 6 highlights problems that standard numerical methods face in reproducing long–run features of the dynamics of the general continuous–time model, while showing these standard methods work in some cases. Chapter 7 develops an alternative method for discretising the solution of the continuous time equation, and shows that it preserves the desired long–run behaviour. Chapter 8 demonstrates that this alternative method converges to the solution of the continuous equation, given sufficient computational effort

New Trends in Differential and Difference Equations and Applications

This is a reprint of articles from the Special Issue published online in the open-access journal Axioms (ISSN 2075-1680) from 2018 to 2019 (available at https://www.mdpi.com/journal/axioms/special issues/differential difference equations)

Perron-Type Criterion for Linear Difference Equations with Distributed Delay

It is shown that if a linear difference equation with distributed delay of the form Δx(n) , n ≥ 1, satisfies a Perron condition then its trivial solution is uniformly asymptotically stable