Cónall Kelly
Lecturer Department
of Mathematics, University of the West
Indies. Room 4, External phone number: +876 977 2693 Extension: 2284/2455 Intercom: 234 Email: conall “dot” kelly “at” uwimona “dot” edu “dot”
jm Research
Stochastic functional differential equations Equilibrium preserving stochastic perturbations
of functional differential equations can have a startling influence on
their qualitative behaviour. Noise and delay have complementary roles
in oscillatory behaviour, and carefully constructed stochastic
perturbations can act to stabilise or destabilise even highly nonlinear
systems with memory effects. I have also worked on the theory of
existence and uniqueness of solutions of nonlinear stochastic
functional differential equations. Discrete-time phenomenological models It can be useful to construct discrete-time
models of stochastic differential equations that capture a particular
dynamic property. For example it is possible to construct stochastic
difference equations that that mimic the oscillatory dynamics of a
particular differential equation, or which reflect the presence or
absence of a finite time explosion in solutions of a corresponding
stochastic differential equation. Stochastic numerical analysis When a numerical method is applied to a
differential equation, the result is a difference equation. Ideally the
dynamics of the difference equation should reflect those of the
original as closely as possible, but in general this can be difficult
to check. It is therefore useful to perform a linear stability
analysis, applying the method of interest to a linear test equation
possessed of an equilibrium solution with known stability properties,
and determining the asymptotic stability of the resultant difference
equation for comparison. Local dynamics in discrete time a.s. asymptotic properties of a class of
polynomial difference equations with several equilibrium solutions,
under the influence of unbounded stochastic perturbation, are local, in
the sense that they depend strongly on the initial state of the
process. This work encroaches upon the field of stochastic numerical
analysis in that these equations may be viewed as Euler-Maruyama
discretisations of drift-polynomial stochastic differential equations
of Ito type. However the resultant dynamic picture holds for more
general perturbation classes, including those with heavy tails. Moreover, one can identify
interesting dynamic features present in discrete time that are
impossible in continuous time. Academic History
My PhD thesis, submitted to Dublin City University in July 2005, is entitled ‘On the Oscillatory
Behaviour of Stochastic Delay Equations’ and you can download the PDF
version here. From August 2005 until August 2006 I held a
postdoctoral position at University College Cork, working with the Probability Group, headed by
Prof. Neil O’Connell. The Probability Group, supported by SFI grant
04/RP1/L512 “Probability and its applications”, was part of the School of Mathematical
Sciences and the Boole Centre for Research
in Informatics. Since August 2006 I have been a lecturer in the
Department of Mathematics
and Computer Science at the University of the West
Indies, Teaching (2008/09)
Semester 1: M65A – Stochastic Processes. Semester 2: M25B - Statistical Inference. M33R - Complex Analysis. Teaching materials for courses currently in
session can be found on OurVLE – the university’s online learning
system, accessible to students here. Seminar Series
I co-ordinate the regular research seminar held in the Mathematics Lecture Theatre on
Fridays at 11am during term.
Publications (preprints and offprints available
on request)
Cónall Kelly and Alexandra Rodkina. Constrained
Stability and Instability of Polynomial Difference Equations with
State-Dependent Noise. Discrete and Continuous Dynamical Systems,
Series A, accepted for publication. John A.D. Appleby, Xuerong Mao, Cónall Kelly,
and Alexandra Rodkina. On
the local stability and instability of polynomial difference equations
with fading stochastic perturbations. Dynamics of Continuous, Discrete and Impulsive
Systems, Series A: Mathematical Analysis, accepted for publication. John A.D. Appleby, Cónall Kelly, and Alexandra
Rodkina. Dynamical consistency of solutions of
continuous and discrete stochastic equations with a finite time
explosion. Proceedings of the
Twelfth International Conference on Difference Equations and
Applications, accepted for publication. John A.D. Appleby, Xuerong Mao, Cónall Kelly,
and Alexandra Rodkina. Positivity and Stabilization for Nonlinear
Stochastic Delay Differential Equations. Stochastics: An International Journal of Probability and Stochastic
Processes. 81 (2009),
no. 1, 29–54. Cónall Kelly and Kirk Morgan. A
Monte-Carlo Approach to the Effect of Noise on Local Stability in
Polynomial Difference Equations. Matematicas: Ensenanza Universitaria. 16 (2008), no. 2, 3-10. John A.D. Appleby and Cónall Kelly. Spurious oscillation in a uniform Euler
discretisation of linear stochastic differential equations with
vanishing delay. Journal of. Computational and Applied
Mathematics. 205 (2007), no. 2, 923-925. John A.D. Appleby and Cónall Kelly. Almost Sure Asymptotic Behaviour of One- and
Two-step Difference Equations with Random Coefficients on a Reducing
Mesh. Proceedings of the IX International Chetayev
Conference: Analytical Mechanics, Stability and Control of Motion, (2007). John A.D. Appleby and Cónall Kelly. Oscillation of solutions of a nonuniform
discretisation of linear stochastic differential equations with
vanishing delay. Dynamics of Continuous, Discrete and Impulsive
Systems, Series A: Mathematical Analysis. 13B (2006), suppl.,
535-550. John A. D. Appleby and
Cónall Kelly. Prevention of Explosions in Solutions of
Functional Differential Equations by Noise Perturbation. Dynamic Systems and Applications,
15
(2006),
no. 2, 227–240. John A. D. Appleby and
Cónall Kelly. Asymptotic and Oscillatory Properties of Linear
Stochastic Delay Differential Equations with Vanishing Delay. Functional Differential Equations,
11 (2004), no. 3–4, 235–265. John A. D. Appleby and
Cónall Kelly. Oscillation and Non-oscillation in Solutions of
Nonlinear Stochastic Delay Differential Equations. Electronic Communications in Probability,
9 (2004), 106–118. Presentations 2008/09
Local
Dynamics of Stochastic Difference Equations, Warwick
Mathematics Institute Seminar, 21
January 2009, Local
Dynamics of Stochastic Difference Equations, Linear
stability analysis of one-step methods for systems of stochastic
differential equations, Local
Convergence of Difference Equations with Fading Gaussian Noise, UWI
Mathematics Seminar, 10
October 2008, University of the West Indies, Local
Convergence of Difference Equations with Fading Gaussian Noise, University
of Puerto Rico Mathematics Seminar, 19
August 2008, Mayaguez, Puerto Rico. Local
Stability of Polynomial Difference Equations Experiencing Unbounded Stochastic
Perturbations, Explosions
and Hard Landings in Discretized Nonlinear Stochastic Equations, IX
International Conference: Approximation and Optimization in the
Caribbean, 6
March 2008, Feedback
Noise in Dynamical Systems, Eighth
Conference: Faculty of Pure and Applied Sciences,
28 February 2008, University of the West Indies, *
Funded by a Return to Academic Staff
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