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John A. D. Appleby

Publications and source records attributed to John A. D. Appleby.

36 records · Page 2Linked to original sources
arXiv2016

Memory Dependent Growth in Sublinear Volterra Differential Equations

We investigate memory dependent asymptotic growth in scalar Volterra equations with sublinear nonlinearity. To obtain precise results we utilise the powerful theory of regular variation extensively. By computing the growth rate in terms of a related ordinary differential equation we show that when the memory effect is so strong that the kernel tends to infinity, the growth rate of solutions depends explicitly on the memory of the system. Finally, we employ a fixed point argument to determine analogous results for a perturbed Volterra equation and show that, for a sufficiently large perturbation, the solution tracks the perturbation asymptotically, even when the forcing term is potentially highly non-monotone.

math.CA↗
arXiv2014

Subexponential Growth Rates in Functional Differential Equations

This paper determines the rate of growth to infinity of a scalar autonomous nonlinear functional differential equation with finite delay, where the right hand side is a positive continuous linear functional of $f(x)$. We assume $f$ grows sublinearly, and is such that solutions should exhibit growth faster than polynomial, but slower than exponential. Under some technical conditions on $f$, it is shown that the solution of the functional differential equation is asymptotic to that of an auxiliary autonomous ordinary differential equation with righthand side proportional to $f$ (with the constant of proportionality equal to the mass of the finite measure associated with the linear functional), provided $f$ grows more slowly than $l(x)=x/\log x$. This linear--logarithmic growth rate is also shown to be critical: if $f$ grows more rapidly than $l$, the ODE dominates the FDE; if $f$ is asymptotic to a constant multiple of $l$, the FDE and ODE grow at the same rate, modulo a constant non--unit factor.

math.CA↗
arXiv2014

On Necessary and Sufficient Conditions for Preserving Convergence Rates to Equilibrium in Deterministically and Stochastically Perturbed Differential Equations with Regularly Varying Nonlinearity

This paper develops necessary and sufficient conditions for the preservation of asymptotic convergence rates of deterministically and stochastically perturbed ordinary differential equations with regularly varying nonlinearity close to their equilibrium. Sharp conditions are also established which preserve the asymptotic behaviour of the derivative of the underlying unperturbed equation. Finally, necessary and sufficient conditions are established which enable finite difference approximations to the derivative in the stochastic equation to preserve the asymptotic behaviour of the derivative of the unperturbed equation, even though the solution of the stochastic equation is nowhere differentiable, almost surely.

math.CA↗
arXiv2013

Exact Pathwise and Mean--Square Asymptotic Behaviour of Stochastic Affine Volterra and Functional Differential Equations

The almost sure rate of exponential-polynomial growth or decay of affine stochastic Volterra and affine stochastic finite-delay equations is investigated. These results are achieved under suitable smallness conditions on the intensities of the deterministic and stochastic perturbations diffusion, given that the asymptotic behaviour of the underlying deterministic resolvent is determined by the zeros of its characteristic equation. The results rely heavily upon a stochastic variant of the admissibility theory for linear Volterra operators.

math.CA↗
arXiv2013

Long Memory and Financial Market Bubble Dynamics in Affine Stochastic Differential Equations with Average Functionals

In this paper we consider the growth, large fluctuations and memory properties of an affine stochastic functional differential equation with an average functional where the contributions of the average and instantaneous terms are parameterised. An asymptotic analysis of the solution of this equation is conducted for all values of the parameters of the equation. When solutions are recurrent, we show that the autocovariance function of the solution decays at a polynomial rate, even though the solution is asymptotically equal to another asymptotically stationary process whose autocovariance function decays exponentially. It is shown that when solutions grow, they do so at either a polynomial or exponential rate in time depending on the sign of a parameter of the model, modulo some exceptional parameter sets. On these exceptional sets, solutions are recurrent on the real line with large fluctuations consistent with the Law of the Iterated logarithm, or exhibit subexponential yet superpolynomial growth.

math.PR↗
arXiv2013

On the Classification of the Asymptotic Behaviour of Solutions of Globally Stable Scalar Differential Equations with Respect to State--Independent Stochastic Perturbations

In this paper we characterise the global stability, global boundedness and recurrence of solutions of a scalar nonlinear stochastic differential equation. The differential equation is a perturbed version of a globally stable autonomous equation with unique equilibrium where the diffusion coefficient is independent of the state. We give conditions which depend on the rate of decay of the noise intensity under which solutions either (a) tend to the equilibrium almost surely, (b) are bounded almost surely but tend to zero with probability zero, (c) or are recurrent on the real line almost surely. We also show that no other types of asymptotic behaviour are possible. Connections between the conditions which characterise the various classes of long--run behaviour and simple sufficient conditions are explored, as well as the relationship between the size of fluctuations and the strength of the mean reversion and diffusion coefficient, in the case when solutions are a.s. bounded.

math.PR↗
arXiv2013

Classification of the Asymptotic Behaviour of Globally Stable Linear Differential Equations with Respect to State-independent Stochastic Perturbations

In this paper we consider the global stability of solutions of an affine stochastic differential equation. The differential equation is a perturbed version of a globally stable linear autonomous equation with unique zero equilibrium where the diffusion coefficient is independent of the state. We find necessary and sufficient conditions on the rate of decay of the noise intensity for the solution of the equation to be globally asymptotically stable, stable but not asymptotically stable, and unstable, each with probability one. In the case of stable or bounded solutions, or when solutions are a.s. unstable asymptotically stable in mean square, it follows that the norm of the solution has zero liminf, by virtue of the fact that $\|X\|^2$ has zero pathwise average a.s. Sufficient conditions guaranteeing the different types of asymptotic behaviour which are more readily checked are developed. It is also shown that noise cannot stabilise solutions, and that the results can be extended in all regards to affine stochastic differential equations with periodic coefficients.

math.PR↗
arXiv2013

Classification of the Asymptotic Behaviour of Globally Stable Differential Equations with Respect to State-independent Stochastic Perturbations

In this paper we consider the global stability of solutions of a nonlinear stochastic differential equation. The differential equation is a perturbed version of a globally stable linear autonomous equation with unique zero equilibrium where the diffusion coefficient is independent of the state. Contingent on a dissipative condition characterising the asymptotic stability of the unperturbed equation, necessary and sufficient conditions on the rate of decay of the noise intensity for the solution of the equation to be a.s. globally asymptotically stable, contingent on some weak and noise independent reversion towards the equilibrium when the solution is far from equilibrium. Under a stronger equilibrium reverting condition, we may classify whether the solution globally asymptotically stable, stable but not asymptotically stable, and unstable, each with probability one purely in terms of the asymptotic intensity of the noise. Sufficient conditions guaranteeing the different types of asymptotic behaviour which are more readily checked are developed.

math.PR↗
arXiv2013

On the almost sure running maxima of solutions of affine neutral stochastic functional differential equations

This paper studies the large fluctuations of solutions of finite--dimensional affine stochastic neutral functional differential equations with finite memory, as well as related nonlinear equations. We find conditions under which the exact almost sure growth rate of the running maximum of each component of the system can be determined, both for affine and nonlinear equations. The proofs exploit the fact that an exponentially decaying fundamental solution of the underlying deterministic equation is sufficient to ensure that the solution of the affine equation converges to a stationary Gaussian process.

math.PR↗
arXiv2013

On the Existence and Uniqueness of Solutions of Stochastic Equations of Neutral Type

This paper considers some the existence and uniqueness of strong solutions of stochastic neutral functional differential equations. The conditions on the neutral functional relax those commonly used to establish the existence and uniqueness of solutions of NSFDEs for many important classes of functional, and parallel the conditions used to ensure existence and uniqueness of solutions of related deterministic neutral equations. Exponential estimates on the almost sure and p--th mean rate of growth of solutions under the weaker existence conditions are also given.

math.PR↗
arXiv2013

On the Dynamic Consistency of the Split Step Method for Classifying the Asymptotic Behaviour of Globally Stable Differential Equations perturbed by State--independent Stochastic terms

In this paper we classify the pathwise asymptotic behaviour of the discretisation of a general autonomous scalar differential equation which has a unique and globally stable equilibrium. The underlying continuous equation is subjected to a stochastic perturbation whose intensity is state--independent. In the main result, it is shown that when the split--step--method is applied to the resulting stochastic differential equation, and the stochastic intensity is decreasing, the solutions of the discretised equation inherit the asymptotic behaviour of the continuous equation, regardless of whether the continuous equation has stable, bounded but unstable, or unbounded solutions, provided the step size is chosen sufficiently small.

math.PR↗
arXiv2012

On the Admissibility of Linear Stochastic Volterra Operators

Conditions guaranteeing convergence of linear stochastic Volterra operators are studied. Necessary and sufficient conditions for mean square convergence are established, while almost sure convergence of the linear operator is shown to imply mean square convergence. Sufficient conditions for almost sure convergence of the stochastic linear operator are established. The sharpness or necessity of these conditions is explored by means of examples.

math.PR↗
arXiv2012

Long run behaviour of the autocovariance function of ARCH($\infty$) models

The asymptotic properties of the memory structure of ARCH($\infty$) equations are investigated. This asymptotic analysis is achieved by expressing the autocovariance function of ARCH($\infty$) equations as the solution of a linear Volterra summation equation and analysing the properties of an associated resolvent equation via the admissibility theory of linear Volterra operators. It is shown that the autocovariance function decays subexponentially (or geometrically) if and only if the kernel of the resolvent equation has the same decay property. It is also shown that upper subexponential bounds on the autocovariance function result if and only if similar bounds apply to the kernel.

math.CA↗
arXiv2012

Necessary and sufficient conditions for periodic decaying resolvents in linear discrete convolution Volterra equations and applications to ARCH$(\infty)$ processes

We define a class of functions which have a known decay rate coupled with a periodic fluctuation. We identify conditions on the kernel of a linear summation convolution Volterra equation which give the equivalence of the kernel lying in this class of functions and the solution lying in this class of functions. Some specific examples are examined. In particular this theory is used to provide a counter--example to a result regarding the rate of decay of the auto--covariance function of an ARCH($\infty$) process.

math.CA↗
arXiv2012

A Black--Scholes Model with Long Memory

This note develops a stochastic model of asset volatility. The volatility obeys a continuous-time autoregressive equation. Conditions under which the process is asymptotically stationary and possesses long memory are characterised. Connections with the class of ARCH($\infty$) processes are sketched.

q-fin.PR↗
arXiv2010

Long Memory in a Linear Stochastic Volterra Differential Equation

In this paper we consider a linear stochastic Volterra equation which has a stationary solution. We show that when the kernel of the fundamental solution is regularly varying at infinity with a log-convex tail integral, then the autocovariance function of the stationary solution is also regularly varying at infinity and its exact pointwise rate of decay can be determined. Moreover, it can be shown that this stationary process has either long memory in the sense that the autocovariance function is not integrable over the reals or is subexponential. Under certain conditions upon the kernel, even arbitrarily slow decay rates of the autocovariance function can be achieved. Analogous results are obtained for the corresponding discrete equation.

math.CA↗