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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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.
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