Cónall Kelly

Towards a Systematic Linear Stability Analysis of Numerical Methods for Systems of Stochastic Differential Equations

We develop two classes of test equations for the linear stability analysis of numerical methods applied to systems of stochastic ordinary differential equations of Ito type (SODEs). Motivated by the theory of stochastic stabilization and destabilization, these test equations capture certain fundamental effects of stochastic perturbation in systems of SODEs, while remaining amenable to analysis before and after discretization. We then carry out a linear stability analysis of the

Almost sure asymptotic stability analysis of the θ -Maruyama method applied to a test system with stabilising and destabilising stochastic perturbations

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Preserving positivity in solutions of discretised stochastic differential equations

-Maruyama method applied to these test equations, investigating mean-square and almost sure asymptotic stability of the test equilibria. We discuss the implications of our work for the notion of A-stability of the

Non-normal drift structures and linear stability analysis of numerical methods for systems of stochastic differential equations

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Stabilisation of difference equations with noisy prediction-based control

-Maruyama method and use numerical simulation to suggest extensions of our results to test systems with nonnormal drift coefficients.

Almost sure instability of the equilibrium solution of a Milstein-type stochastic difference equation

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. For small values of the constant step-size parameter, we derive close-to-sharp conditions for the almost sure asymptotic stability and instability of the equilibrium solution of the discretisation that match those of the original test system. Our investigation demonstrates the use of a discrete form of the Itô formula in the context of an almost sure linear stability analysis.

Asymptotic and Transient Mean-Square Properties of Stochastic Systems Arising in Ecology, Fluid Dynamics, and System Control

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.

Positivity and stabilisation for nonlinear stochastic delay differential equations

We investigate mean-square asymptotic stability of equilibria in linear systems of stochastic differential equations with non-normal drift coefficients, with particular emphasis on the role of interactions between the drift and diffusion structures that act along, orthogonally to, and laterally to the flow. Hence we construct test systems with non-normal drift coefficients and characteristic diffusion structures for the purposes of a linear stability analysis of the @q-Maruyama method. Next we discretise these test systems and examine the mean-square asymptotic stability of equilibria of the resulting systems of stochastic difference equations. Finally we indicate how this approach may help to shed light on numerical discretisations of stochastic partial differential equations with multiplicative space-time perturbations.

Stochastic stability analysis of a reduced galactic dynamo model with perturbed α-effect

Abstract We consider the influence of stochastic perturbations on stability of a unique positive equilibrium of a difference equation subject to prediction-based control. These perturbations may be multiplicative x n + 1 = f ( x n ) − ( α + l ξ n + 1 ) ( f ( x n ) − x n ) , n = 0 , 1 , … , if they arise from stochastic variation of the control parameter, or additive x n + 1 = f ( x n ) − α ( f ( x n ) − x n ) + l ξ n + 1 , n = 0 , 1 , … , if they reflect the presence of systemic noise. We begin by relaxing the control parameter in the deterministic equation, and deriving a range of values for the parameter over which all solutions eventually enter an invariant interval. Then, by allowing the variation to be stochastic, we derive sufficient conditions (less restrictive than known ones for the unperturbed equation) under which the positive equilibrium will be globally a.s. asymptotically stable: i.e. the presence of noise improves the known effectiveness of prediction-based control. Finally, we show that systemic noise has a “blurring” effect on the positive equilibrium, which can be made arbitrarily small by controlling the noise intensity. Numerical examples illustrate our results.

Adaptive timestepping for pathwise stability and positivity of strongly discretised nonlinear stochastic differential equations

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 Ito stochastic differential equation. Our analysis relies upon the convergence of non-negative martingale sequences coupled with a discrete form of the Ito 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.