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Classication of the asymptotic behaviour of solutions of stochastic differential equations with state independent noise

Cheng, Jian (2012) Classication of the asymptotic behaviour of solutions of stochastic differential equations with state independent noise. PhD thesis, Dublin City University.

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Abstract

We investigate the asymptotic behaviour of solution of dierential equation with state-independent perturbation. The dierential equation studied is a perturbed version of a globally stable autonomous equation with unique equilibrium where the diffusion coefficient is independent of the state. Perturbed differential equation is widely applied to model natural phenomena, in Finance, Engineering, Physics and other disciplines. Real-world processes are often subjected to interference in the form of random external perturbations. This could lead to a dramatic effect on the behaviour of these processes. Therefore it is important to analyse these equations. We start by considering an additive deterministic perturbation in Chapter 1. It is assumed that the restoring force is asymptotically negligible as the solution becomes large, and that the perturbation tends to zero as time becomes indefinitely large. It is shown that solutions are always locally stable, and that solutions either tend to zero or to infinity as time tends to infinity. In Chapter 2 and 4, we each explore a linear and nonlinear equation with stochastic perturbation in finite dimensions. We find necessary and sufficient conditions on the rate of decay of the noise intensity for the solution of the equations to be globally asymptotically stable, bounded, or unstable. In Chapter 3 we concentrate on a scalar nonlinear stochastic differential equation. As well as the necessary and sufficient condition, we also explore the simple sufficient conditions and the connections between the conditions which characterise the various classes of long-run behaviour. To facilitate the analysis, we investigate using Split-Step method the difference equations both in the scalar case and the finite dimensional case in Chapter 5 and 6. We can mimic the exact asymptotic behaviour of the solution of the stochastic differential equation under the same conditions in discrete time.

Item Type:Thesis (PhD)
Date of Award:November 2012
Refereed:No
Supervisor(s):Appleby, John
Uncontrolled Keywords:Stochastic differential equations; split step Euler method; almost sure asymptotic stability; state independent perturbation; almost sure global stability; boundedness
Subjects:Mathematics > Dynamics
Mathematics > Differential equations
Mathematics > Numerical analysis
Mathematics
Mathematics > Stochastic analysis
Mathematics > Mathematical analysis
DCU Faculties and Centres:DCU Faculties and Schools > Faculty of Science and Health > School of Mathematical Sciences
Use License:This item is licensed under a Creative Commons Attribution-NonCommercial-No Derivative Works 3.0 License. View License
Funders:Science Foundation Ireland
ID Code:17501
Deposited On:29 Nov 2012 14:25 by John Appleby. Last Modified 19 Mar 2013 04:02

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