Books

• Numerical Methods for Finance, John Miller, David Edelman and John Appleby, 312pp, Chapman and Hall/CRC, 2007, ISBN-13: 978-1584889250
• Catherine Swords and John Appleby, ``Stochastic Delay Difference and Differential Equations: Applications to Financial Markets'', 180pp, LAP Lambert Academic Publishing, 2010, ISBN-13: 978-3838334752.
• Huizhong Wu and John Appleby, ``Pathwise Large Deviations of Stochastic Differential Equations with Applications to Finance'', 200pp, LAP LAMBERT Academic Publishing, 2010, ISBN-13: 978-3838360447.
• Terry Lynch and John Appleby, ``Large Fluctuations of Stochastic Differential Equations: Regime Switching and Applications to Simulation and Finance'', 240pp, LAP LAMBERT Academic Publishing, 2010, ISBN-13: 978-3843359351.

Refereed Publications

• Heterogeneity and seasonality in Financial Markets, International Journal of Theoretical and Applied Finance, 3(3), 491--2, 2000.
• A Complete Market Model with Feedback, Proceedings, 16th IMACS World Congress, 6pp, 2000.
• On the non-exponential convergence of asymptotically stable solutions of linear Volterra integro-differential equations (with David Reynolds), J. Integral Equations Appl., 14(2), 109--118, 2002.
• Subexponential solutions of linear integro-differential equations and transient renewal equations (with David Reynolds), Proc. Royal Soc. Edinburgh, 132A, 521--543, 2002.
• Almost sure stability of linear Itô-Volterra equations with damped stochastic perturbations, Electron. Commun. Probab., 7, Paper no. 22, 223--234, 2002.
• Non-exponential stability of scalar stochastic Volterra equations (with D. Reynolds), Statist. Probab. Lett., 62(4), 335--343, 2003.
• Exponential asymptotic stability of linear Itô-Volterra equations with damped stochastic perturbations, (with Alan Freeman), Electron. J. Probab., 8, Paper No. 22, 22pp, 2003.
• Corrigendum "Subexponential solutions of linear Volterra integro-differential equations and transient renewal equations" (with David Reynolds). Proc. Roy. Soc. Edinburgh. Sect. A , 133: 1421, 2003.
• p-th-mean integrability and almost sure asymptotic stability of Itô-Volterra equations, J. Integral Equations Appl., 15(4), 321--341, 2003.
• Subexponential solutions of scalar linear Itô-Volterra equations with damped stochastic perturbations, Funct. Differ. Equ., 11(1-2), 5--10, 2004.
• Subexponential solutions of linear scalar integrodifferential equations with a delay (with David Reynolds, István Gyõri), Funct. Differ. Equ. 11(1-2), 11--18, 2004.
• Almost sure subexponential decay rates of scalar Itô-Volterra equations, Electron. J. Qual. Theory Differ. Equ., Proc. 7th Coll. QTDE, 2004, No.1, 1--32.
• On the nonexponential decay to equilibrium of solutions of nonlinear scalar Volterra integro-differential equations (with David Reynolds), Electron. J. Qual. Theory Differ. Equ., Proc. 7th Coll. QTDE, 2004, No.3, 1--14.
• Almost sure polynomial asymptotic stability of scalar stochastic differential equations with damped stochastic perturbations (with Dana Mackey) Electron. J. Qual. Theory Differ. Equ., Proc. 7th Coll. QTDE, 2004, No.2, 1-33.
• Oscillation and non-oscillation in solutions of nonlinear stochastic delay differential equations (with Cónall Kelly), Electron. Comm. Probab., 9, 106--118, 2004.
• On necessary and sufficient conditions for exponential stability in linear Volterra integro--differential equations (with David Reynolds), J. Integral Equations Appl., 16(3), 221--240, 2004.
• Subexponential solutions of linear integrodifferential equations (with David Reynolds), Proceedings Dynam. Systems Appl. 4, 488--494, 2004.
• Asymptotic and oscillatory properties of linear stochastic delay differential equations with vanishing delay, (with Cónall Kelly) Funct. Differ. Eqn., 11(3-4), 235--265, 2004.
• Decay and growth rates of solutions of scalar stochastic delay differential equations with unbounded delay and state dependent noise, Stochastics and Dynamics, 5(2), 133--148, 2005.
• On the asymptotic stability of polynomial stochastic delay differential equations (with Alexandra Rodkina), Funct. Differ. Equ., 12(1--2), 35--66, 2005.
• Rates of decay and growth of solutions to linear stochastic differential equations with state--independent perturbations (with Alexandra Rodkina), Stochastics, 77(3), 271--295, 2005.
• Pathwise superexponential decay rates of solutions of autonomous stochastic differential equations (with Xuerong Mao, Alexandra Rodkina), Stochastics, 77(3), 241--270, 2005.
• Exponential convergence to a nontrivial limit in linear Volterra integrodifferential equations (with Siobhán Devin, David Reynolds), Electron. J. Qual. Th. Differ. Equ, No. 9. (2005), pp. 1-16.
• Exponential asymptotic stability of nonlinear Itô-Volterra equations with damped stochastic perturbations, Funct. Differ. Equ., 12(1--2), 7--34, 2005.
• Noise induced oscillation in solutions of stochastic delay differential equations (with Evelyn Buckwar), Dynam. Systems Appl., 14(2), 175--196, 2005.
• Stochastic Stabilisation of Functional Differential Equations (with Xuerong Mao), System Control Lett., 54, 1069--1081, 2005.
• On stochastic stabilization of difference equations (with Xuerong Mao, Alexandra Rodkina), Disc. Cont. Dynam. Sys. A, 15(3), 843--857, 2006.
• Pathwise non-exponential decay rates of solutions of scalar nonlinear stochastic differential equation, (with Alexandra Rodkina, Henri Schurz), Disc. Con. Dynam. Sys. Ser. B., 6(4), 667--696, 2006. %See {\bf[S18]}).
• Prevention of Explosions in Solutions of Functional Differential Equations by Noise Perturbation (with Cónall Kelly) Dynam. Systems Appl., 15(2), 227--240, 2006.
• On exact rates of decay of solutions of linear systems of Volterra equations with delay (with David Reynolds, István Gyõri), J. Math. Anal. Appl., 320, 56--77, 2006.
• Almost sure asymptotic stability of stochastic Volterra integro--differential equations with fading perturbations (with Markus Riedle), Stochastic Anal. Appl., 24(4), 813--826, 2006.
• On Exponential Asymptotic Stability in Linear Viscoelasticity (with Mauro Fabrizio, Barbara Lazzari, David W. Reynolds), Mathematical Modelling in Engineering, 16(10), 1677--1694, 2006.
• Almost sure polynomial asymptotic stability of stochastic difference equations (with Dana Mackey, Alexandra Rodkina), Contemporary Mathematics: Fundamental Directions, 17, 110--128, 2006 (in Russian); Journal of Mathematical Sciences, 149, Number 6, 1629-1647, 2008 (in English).
• Mean square convergence of solutions of linear Itô-Volterra equations to non--equilibrium limits (with Siobhán Devin and David Reynolds), Dynam. Con. Disc. Imp. Sys. Ser A Math Anal., 13B, suppl., 515--534, 2006.
• Oscillation of solutions of a nonuniform discretisation of linear stochastic differential equations with vanishing delay (with Cónall Kelly), Dynam. Con. Disc. Imp. Sys. Ser A Math Anal., 13B, suppl., 535--550, 2006.
• On exact convergence rates for solutions of linear systems of Volterra difference equations (with David Reynolds, István Gyori), J. Differ. Equ. Appl., 12(12), 1257-1275, 2006. Corrigendum: 13(1), 2007, p. 95.
• Stabilisation of Volterra equations by noise, (with Aoife Flynn), J. Appl. Math. Stoch. Anal., 29pp, Art. ID 89729, 2006.
• On the asymptotic convergence to a non--equilibrium limit of solutions of linear stochastic Volterra equations (with Siobhán Devin, David Reynolds), J. Integral Equations Appl., 19(4), 405--437, 2007.
• Spurious oscillation in a uniform Euler discretisation of linear stochastic differential equations with vanishing delay (with Cónall Kelly), J. Comput. Appl. Math., 205 (2), 923--935, 2007.
• On exact rates of growth and decay of solutions of a linear Volterra equation in linear viscoelasticity (with István Gy\H{o}ri, David Reynolds), \Note de Matematica, 212--226, 2007.
• On polynomial Rate of Decay and Growth of Solution to Nonlinear Difference Equations (with Ma\l gorzata Guzowska, Alexandra Rodkina, Rev. Mate. Teor. Aplic., 14(2), 33--41, 2007.
• On the asymptotic behavior of the moments of solutions of stochastic difference equations (with A.~Rodkina, G.~Berkoliako), 557--565, in: Difference Equations, Special Functions and Orthogonal Polynomials: Proceedings of the International Conference. Editors: S.~Elaydi, J.~Cushing, R.~Lasser, A.~Ruffing, V.~Papageorgiou \& W.~Van Assche, World Scientific, 2007.
• On the Oscillation of Solutions of Stochastic Difference Equations with State-Independent Perturbations (with Alexandra Rodkina), Intern. J. Difference Equ., 2 (2), 139–-164, 2007.
• Almost Sure Asymptotic Behaviour of One- and Two-step Difference Equations with Random Coefficients on a Reducing Mesh (with Cónall Kelly). In: Proceedings of the Eleventh International Chetayev Conference: Analytical Mechanics, Stability and Control of Motion, Institute for System Dynamics and Control Theory of Siberian Branch of the Russian Academy of Science, Irkutsk, Russia. Volume 5, pp14, 2007.
• Decay rates of solutions of linear stochastic Volterra equations, (with David Reynolds), Electron. J. Probab., 13, Paper. 30, 922--943, 2008.
• Stabilization and Destabilization of Nonlinear Differential Equations by Noise (with Xuerong Mao, Alexandra Rodkina), IEEE Transactions in Automatic Control , 53 (3), 683--691, 2008.
• Fat Tails and Bubbles in a Discrete Time Model of an Inefficient Financial Market, (with A. Rodkina, C. Swords), Proceedings Dynam. Systems Appl. 5, 35--45, 2008.
• Asymptotic constancy and stability in nonautonomous stochastic differential equations (with Alexandra Rodkina, James Gleeson) Cubo J., 10 (3), 145--160, 2008.
• Characterisation of exponential convergence to non--equilibrium limits for stochastic Volterra equations (with Siobhán Devin, David Reynolds J. Appl. Math. Stoch. Anal., 2008, Article ID 473156, 27pp, 2008.
• Exponential growth and Gaussian--like fluctuations of solutions of stochastic differential equations with maximum functionals (with Huizhong Wu), J. Physics: Conference Series, 138, 2008, paper number 012002, 25pp.
• Convergence rates of the solution of a Volterra--type stochastic differential equations to a non--equilibrium limit, Electron. J. Qual. Theory Differ. Equ., Proc. 8th Coll. QTDE, 2008, No. 1., 1-30.
• On Local Stability for a Nonlinear Difference Equation with a Non-Hyperbolic Equilibrium and Fading Stochastic Perturbations (with G.Berkolaiko, A. Rodkina) J. Differ. Equ. Appl., 14 (9), 923--951, 2008.
• Geometric Brownian Motion with delay: mean square characterisation (with Xuerong Mao, Markus Riedle), Proc. Amer. Math. Soc., 137, 339--348, 2009.
• Asymptotic behaviour of a nonlinear stochastic difference equation modelling an inefficient financial market (with Catherine Swords), Adv. Stud. Pure Math., 53, 23--33, 2009.
• Growth Rates of Delay--Differential Equations and Uniform Euler Schemes (with M. McCarthy, A. Rodkina), in "Difference Equations and Applications", Proceedings of the 14th International Conference on Difference Equations and Applications, pp 117--124, 2009.
• Positivity and stabilization for nonlinear stochastic delay differential equations (with Cónall Kelly, Xuerong Mao, Alexandra Rodkina) Stochastics, 81, 1, 29--54, 2009.
• Nonexponential stability and Decay Rates of Nonlinear Stochastic Difference Equations with unbounded noises (with Alexandra Rodkina, Gregory Berkolaiko), Stochastics, 81, 2, 99-–127, 2009.
• Solutions of stochastic differential equations obeying the law of the iterated logarithm with applications to financial markets (with H.Wu), Electron. J. Probab, 14, 912--959, 2009.
• The Size of the Largest Fluctuations in a Market Model with Markovian Switching (with Terry Lynch, Xuerong Mao, Huizhong Wu, Commun. Appl. Anal., 13 (2), 135-–166, 2009.
• On asymptotic stability and instability with respect to a fading stochastic perturbation (with James Gleeson, Alexandra Rodkina), Applicable Analysis, 88 (4), 579-–603, 2009.
• On Asymptotic Stability of linear stochastic Volterra difference equations with respect to a fading perturbation (with Alexandra Rodkina, Markus Riedle) Adv. Stud. Pure Math., 53, 271--282, 2009.
• Stability of a limit cycle for a planar system with stochastic perturbations (with Alexandra Rodkina and Lih--Ing W. Roeger), Funct. Differ. Equ., 16 (1), 73--91, 2009.
• Stability of Nonlinear Stochastic Volterra Difference Equations with Respect to a Fading Perturbation, (with Alexandra Rodkina) International Journal of Difference Equations, 4 (2), 165-–184, 2009.
• On the oscillation of solutions of stochastic difference equations (with Alexandra Rodkina and Henri Schurz), 17 (2), Matemáticas: Enseñanza Universitaria, 1–10, 2009.
• Stochastic Volterra differential equations in weighted spaces (with Markus Riedle), J. Integral Equ. Appl., 22 (1), 1--17, 2010.
• History--Dependent and Regularly Varying Convergence Rates of Solutions of Infinite Memory Volterra Integrodifferential Equations, Adv. Differ. Equ., Volume 2010, Article ID 478291, 31 pp.
• On the Positivity and Zero Crossings of Solutions of Stochastic Volterra Integrodifferential Equations, Int. J. Differ. Equ., Volume 2010 (2010), Article ID 508217, 25 pp.
• On the almost sure partial maxima of solutions of affine stochastic functional differential equations (with X.Mao and H.Wu), SIAM J. Math. Analysis , 42(2), 646-–678, 2010.
• On the local stability and instability of polynomial difference equations with fading stochastic perturbations (with Xuerong Mao, Cónall Kelly, and Alexandra Rodkina), Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis, 17, 401--430, 2010.
• Exact growth rates of solutions of delay--dominated differential equations, (with Michael J. McCarthy and Alexandra Rodkina), Proc. Neural, Parallel Scientific Computations IV, 37-42, 2010.
• The split-step Euler--Maruyama method preserves asymptotic stability for simulated annealing problems (with J. Cheng, A. Rodkina), Proc. Neural, Parallel Scientific Computations IV, 31--36, 2010.
• Non-positivity and oscillations of solutions of nonlinear stochastic difference equations with state-dependent noise (with Alexandra Rodkina and Henri Schurz), J. Differ. Equ. Appl., 16(7), 807--830, 2010.
• History--dependent decay rates for a logistic equation with infinite delay (with David Reynolds and István Gyori), Proceedings of Royal Society Edinburgh, A, 141 (3), 23--44, 2011.
• Preservation of positivity in the solution of discretised stochastic differential equation (with Ma\l gorzata Guzowska, Alexandra Rodkina, C. Kelly) Applied Mathematics and Computation, 217 (2), 763--774, 2010.
• On the Use of Adaptive Meshes to Counter Overshoot in Solutions of Discretised Nonlinear Stochastic Differential Equations, (with Alexandra Rodkina, Cónall Kelly) International J. Difference Equ., 5(2), 2010, 129–-148.
• Exponential growth in the solution of an affine stochastic differential equation with an average functional (with J. Daniels), Disc. Continuous. Dynam. Systems, Supplement, 91--101, 2011.
• Characterisation of the asymptotic behaviour of scalar linear differential equations with respect to a fading stochastic perturbation (with A. Rodkina, J. Cheng), Disc. Continuous. Dynam. Systems, Supplement, 79--90, 2011.
• A constructive comparison technique for determining the asymptotic behaviour of linear functional differential equations with unbounded delay, (with Evelyn Buckwar), Differential Equations and Dynamical Systems , 18 (3), 271--301, 2010.
• Long Memory in a Linear Stochastic Volterra Differential Equation (with K. Krol). J. Math. Anal. Appl, 380 (2), 814-830, 2011.
• On an exponential martingale approach to almost sure stability of Itô SDEs in R^1 (with Alexandra Rodkina, Henri Schurz), Dyn. Contin. Discrete Impuls. Syst., Ser. A, Math. Anal. 18 (4), 471--484, 2011.
• Dynamical consistency of solutions of continuous and discrete stochastic equations with a finite time explosion (with Cónall Kelly and Alexandra Rodkina), Elaydi, Saber (ed.) et al., Discrete dynamics and difference equations. Proceedings of the twelfth international conference on difference equations and applications (ICDEA), Lisbon, Portugal, July 23--27, 2007. Hackensack, NJ: World Scientific, 163--172, 2010.
• Non--exponential stability and decay rates in nonlinear stochastic homogeneous difference equations (with G.Berkolaiko, A.Rodkina), Elaydi, Saber (ed.) et al., Discrete dynamics and difference equations. Proceedings of the twelfth international conference on difference equations and applications (ICDEA), Lisbon, Portugal, July 23--27, 2007. Hackensack, NJ: World Scientific, 155--162, 2010.