It is possible to develop a partial description of the oscillatory behaviour of a linear stochast... more It is possible to develop a partial description of the oscillatory behaviour of a linear stochastic differential equation with vanishing delay, by applying a transformation that yields a process with identical oscillatory behaviour, and differentiable sample paths. Results from deterministic theory can then be applied on a pathwise basis. If it is possible to construct a discrete process that mimics
The existence and uniqueness of global solutions of a class of scalar stochastic functional diere... more The existence and uniqueness of global solutions of a class of scalar stochastic functional dierential equations of Ito type is studied. It is not assumed, however, that the coecients need to satisfy global linear bounds. For a subclass of these equations, it is known that the associated deterministic equation, which is not noise-perturbed, explodes in finite time. Therefore, a noise
This paper is concerned with the asymptotic and oscillatory proper- ties of stochastic delay dier... more This paper is concerned with the asymptotic and oscillatory proper- ties of stochastic delay dierential
Discrete Dynamics and Difference Equations - Proceedings of the Twelfth International Conference on Difference Equations and Applications, 2010
163 Dynamical Consistency of Solutions of Continuous and Discrete Stochastic Equations with a Fin... more 163 Dynamical Consistency of Solutions of Continuous and Discrete Stochastic Equations with a Finite Time Explosion John AD Appleby, Cónall Kelly∗ and Alexandra Rodkina∗ School of Mathematical Sciences, Dublin City ... This question is addressed in Dávila et al. ...
We construct discrete models that mimic the asymptotic behaviour of a linear stochastic vanishing... more We construct discrete models that mimic the asymptotic behaviour of a linear stochastic vanishing delay equation in circumstances where the solutions do not oscillate.
We identify putative load-bearing structures (bridges) in experimental colloidal systems studied ... more We identify putative load-bearing structures (bridges) in experimental colloidal systems studied by confocal microscopy. Bridges are co-operative structures that have been used to explain stability and inhomogeneous force transmission in simulated granular packings ...
We present an analysis of the stability behaviour of a class of one-step difference equations des... more We present an analysis of the stability behaviour of a class of one-step difference equations describing an iterated polynomial mapping. Such equations are commonly used to model population dynamics in discrete time. We use Monte-Carlo methods to investigate the effect of a state-dependent random perturbation on the local stability of such equations. In particular we focus on the probability of stability in transitionary initial-value regions; regions where a switch in the qualitative behaviour of the deterministic equation is observed.
We consider two classes of nonlinear stochastic differential equation with a.s. positive solution... more We consider two classes of nonlinear stochastic differential equation with a.s. positive solutions. In the first case the drift coefficient is strongly zero-reverting, and dominates the diffusion, whereas in the second the diffusion is highly variable and dominates the drift. In each case, the tendency to overshoot zero prevents a uniform Euler discretisation from preserving positivity in solutions. To address this, we construct adaptive meshes allowing the generation of positive trajectories with arbitrarily high probability. For completeness, we generalise the analysis to finite-dimensional systems of stochastic differential equations, investigating the effect of a uniform Euler discretisation on the positivity of systems with coefficients satisfying linear bounds, and introducing an adaptive mesh to counter overshoot when those bounds are violated.
We derive a condition guaranteeing the almost sure instability of the equilibrium of a stochastic... more We derive a condition guaranteeing the almost sure instability of the equilibrium of a stochastic difference equation with a structure motivated by the Euler-Milstein discretisation of an Itô stochastic differential equation. Our analysis relies upon the convergence of non-negative martingale sequences coupled with a discrete form of the Itô formula and requires a distinct variant of this formula for each of the linear and nonlinear cases. The conditions developed in this article appear to be quite sharp.
Stochastics An International Journal of Probability and Stochastic Processes, 2009
We use state dependent Gaussian perturbations to stabilise the solutions of differential equation... more We use state dependent Gaussian perturbations to stabilise the solutions of differential equations with coefficients that take, as arguments, averaged sets of information from the history of the solution, as well as isolated past and present states. The properties that guarantee stability also guarantee positivity of solutions as long as the initial value is nonzero.
ABSTRACT In the original article [LMS J. Comput. Math. 15, 71–83 (2012; Zbl 06316291)], the autho... more ABSTRACT In the original article [LMS J. Comput. Math. 15, 71–83 (2012; Zbl 06316291)], the authors use a discrete form of the Itô formula, developed by J. A. D. Appleby et al. [Stochastics 81, No. 2, 99–127 (2009; Zbl 1177.39020)], to show that the almost sure asymptotic stability of a particular two-dimensional test system is preserved when the discretisation step size is small. In this Corrigendum, we identify an implicit assumption in the original proof of the discrete Itô formula that, left unaddressed, would preclude its application to the test system of interest. We resolve this problem by reproving the relevant part of the discrete Itô formula in such a way that confirms its applicability to our test equation. Thus, we reaffirm the main results and conclusions of the original article.
We perform an almost sure linear stability analysis of the θ-Maruyama method, selecting as our te... more We perform an almost sure linear stability analysis of the θ-Maruyama method, selecting as our test equation a two-dimensional system of Itô differential equations with diagonal drift coefficient and two independent stochastic perturbations which capture the stabilising and destabilising roles of feedback geometry in the almost sure asymptotic stability of the equilibrium solution.
Journal of Computational and Applied Mathematics, 2007
It is possible to develop a partial description of the oscillatory behaviour of a linear stochast... more It is possible to develop a partial description of the oscillatory behaviour of a linear stochastic differential equation with vanishing delay, by applying a transformation that yields a process with identical oscillatory behaviour, and differentiable sample paths. Results from deterministic theory can then be applied on a pathwise basis. If it is possible to construct a discrete process that mimics the behaviour of the continuous process, the analysis of that behaviour may be more straightforward, and the resulting description more complete. However, a uniform Euler discretisation yields spurious oscillatory behaviour. Here, an analysis of this behaviour is presented, and an alternative method suggested.
We consider the Euler discretisation of a scalar linear test equation with positive solutions and... more We consider the Euler discretisation of a scalar linear test equation with positive solutions and show for both strong and weak approximations that the probability of positivity over any finite interval of simulation tends to unity as the step size approaches zero. Although a.s. positivity in an approximation is impossible to achieve, we develop for the strong (Maruyama) approximation an asymptotic estimate of the number of mesh points required for positivity as our tolerance of non-positive trajectories tends to zero, and examine the effectiveness of this estimate in the context of practical numerical simulation. We show how this analysis generalises to equations with a drift coefficient that may display a high level of nonlinearity, but which must be linearly bounded from below (i.e. when acting towards zero), and a linearly bounded diffusion coefficient. Finally, in the linear case we develop a refined asymptotic estimate that is more useful as an a priori guide to the number of mesh points required to produce positive approximations with a given probability.
equations with coecients that take, as arguments, averaged sets of information from the history o... more equations with coecients that take, as arguments, averaged sets of information from the history of the solution, as well as isolated past and present states. The properties that guarantee stability also guarantee positivity of solutions as long as the initial value is nonzero. We do not require that any component of the coecients of the equations satisfy Lipschitz conditions. Instead,
It is possible to develop a partial description of the oscillatory behaviour of a linear stochast... more It is possible to develop a partial description of the oscillatory behaviour of a linear stochastic differential equation with vanishing delay, by applying a transformation that yields a process with identical oscillatory behaviour, and differentiable sample paths. Results from deterministic theory can then be applied on a pathwise basis. If it is possible to construct a discrete process that mimics
The existence and uniqueness of global solutions of a class of scalar stochastic functional diere... more The existence and uniqueness of global solutions of a class of scalar stochastic functional dierential equations of Ito type is studied. It is not assumed, however, that the coecients need to satisfy global linear bounds. For a subclass of these equations, it is known that the associated deterministic equation, which is not noise-perturbed, explodes in finite time. Therefore, a noise
This paper is concerned with the asymptotic and oscillatory proper- ties of stochastic delay dier... more This paper is concerned with the asymptotic and oscillatory proper- ties of stochastic delay dierential
Discrete Dynamics and Difference Equations - Proceedings of the Twelfth International Conference on Difference Equations and Applications, 2010
163 Dynamical Consistency of Solutions of Continuous and Discrete Stochastic Equations with a Fin... more 163 Dynamical Consistency of Solutions of Continuous and Discrete Stochastic Equations with a Finite Time Explosion John AD Appleby, Cónall Kelly∗ and Alexandra Rodkina∗ School of Mathematical Sciences, Dublin City ... This question is addressed in Dávila et al. ...
We construct discrete models that mimic the asymptotic behaviour of a linear stochastic vanishing... more We construct discrete models that mimic the asymptotic behaviour of a linear stochastic vanishing delay equation in circumstances where the solutions do not oscillate.
We identify putative load-bearing structures (bridges) in experimental colloidal systems studied ... more We identify putative load-bearing structures (bridges) in experimental colloidal systems studied by confocal microscopy. Bridges are co-operative structures that have been used to explain stability and inhomogeneous force transmission in simulated granular packings ...
We present an analysis of the stability behaviour of a class of one-step difference equations des... more We present an analysis of the stability behaviour of a class of one-step difference equations describing an iterated polynomial mapping. Such equations are commonly used to model population dynamics in discrete time. We use Monte-Carlo methods to investigate the effect of a state-dependent random perturbation on the local stability of such equations. In particular we focus on the probability of stability in transitionary initial-value regions; regions where a switch in the qualitative behaviour of the deterministic equation is observed.
We consider two classes of nonlinear stochastic differential equation with a.s. positive solution... more We consider two classes of nonlinear stochastic differential equation with a.s. positive solutions. In the first case the drift coefficient is strongly zero-reverting, and dominates the diffusion, whereas in the second the diffusion is highly variable and dominates the drift. In each case, the tendency to overshoot zero prevents a uniform Euler discretisation from preserving positivity in solutions. To address this, we construct adaptive meshes allowing the generation of positive trajectories with arbitrarily high probability. For completeness, we generalise the analysis to finite-dimensional systems of stochastic differential equations, investigating the effect of a uniform Euler discretisation on the positivity of systems with coefficients satisfying linear bounds, and introducing an adaptive mesh to counter overshoot when those bounds are violated.
We derive a condition guaranteeing the almost sure instability of the equilibrium of a stochastic... more We derive a condition guaranteeing the almost sure instability of the equilibrium of a stochastic difference equation with a structure motivated by the Euler-Milstein discretisation of an Itô stochastic differential equation. Our analysis relies upon the convergence of non-negative martingale sequences coupled with a discrete form of the Itô formula and requires a distinct variant of this formula for each of the linear and nonlinear cases. The conditions developed in this article appear to be quite sharp.
Stochastics An International Journal of Probability and Stochastic Processes, 2009
We use state dependent Gaussian perturbations to stabilise the solutions of differential equation... more We use state dependent Gaussian perturbations to stabilise the solutions of differential equations with coefficients that take, as arguments, averaged sets of information from the history of the solution, as well as isolated past and present states. The properties that guarantee stability also guarantee positivity of solutions as long as the initial value is nonzero.
ABSTRACT In the original article [LMS J. Comput. Math. 15, 71–83 (2012; Zbl 06316291)], the autho... more ABSTRACT In the original article [LMS J. Comput. Math. 15, 71–83 (2012; Zbl 06316291)], the authors use a discrete form of the Itô formula, developed by J. A. D. Appleby et al. [Stochastics 81, No. 2, 99–127 (2009; Zbl 1177.39020)], to show that the almost sure asymptotic stability of a particular two-dimensional test system is preserved when the discretisation step size is small. In this Corrigendum, we identify an implicit assumption in the original proof of the discrete Itô formula that, left unaddressed, would preclude its application to the test system of interest. We resolve this problem by reproving the relevant part of the discrete Itô formula in such a way that confirms its applicability to our test equation. Thus, we reaffirm the main results and conclusions of the original article.
We perform an almost sure linear stability analysis of the θ-Maruyama method, selecting as our te... more We perform an almost sure linear stability analysis of the θ-Maruyama method, selecting as our test equation a two-dimensional system of Itô differential equations with diagonal drift coefficient and two independent stochastic perturbations which capture the stabilising and destabilising roles of feedback geometry in the almost sure asymptotic stability of the equilibrium solution.
Journal of Computational and Applied Mathematics, 2007
It is possible to develop a partial description of the oscillatory behaviour of a linear stochast... more It is possible to develop a partial description of the oscillatory behaviour of a linear stochastic differential equation with vanishing delay, by applying a transformation that yields a process with identical oscillatory behaviour, and differentiable sample paths. Results from deterministic theory can then be applied on a pathwise basis. If it is possible to construct a discrete process that mimics the behaviour of the continuous process, the analysis of that behaviour may be more straightforward, and the resulting description more complete. However, a uniform Euler discretisation yields spurious oscillatory behaviour. Here, an analysis of this behaviour is presented, and an alternative method suggested.
We consider the Euler discretisation of a scalar linear test equation with positive solutions and... more We consider the Euler discretisation of a scalar linear test equation with positive solutions and show for both strong and weak approximations that the probability of positivity over any finite interval of simulation tends to unity as the step size approaches zero. Although a.s. positivity in an approximation is impossible to achieve, we develop for the strong (Maruyama) approximation an asymptotic estimate of the number of mesh points required for positivity as our tolerance of non-positive trajectories tends to zero, and examine the effectiveness of this estimate in the context of practical numerical simulation. We show how this analysis generalises to equations with a drift coefficient that may display a high level of nonlinearity, but which must be linearly bounded from below (i.e. when acting towards zero), and a linearly bounded diffusion coefficient. Finally, in the linear case we develop a refined asymptotic estimate that is more useful as an a priori guide to the number of mesh points required to produce positive approximations with a given probability.
equations with coecients that take, as arguments, averaged sets of information from the history o... more equations with coecients that take, as arguments, averaged sets of information from the history of the solution, as well as isolated past and present states. The properties that guarantee stability also guarantee positivity of solutions as long as the initial value is nonzero. We do not require that any component of the coecients of the equations satisfy Lipschitz conditions. Instead,
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Papers by Cónall Kelly