# Non-exponential stability and decay rates in nonlinear stochastic difference equations with unbounded noise

@article{Appleby2009NonexponentialSA, title={Non-exponential stability and decay rates in nonlinear stochastic difference equations with unbounded noise}, author={John A. D. Appleby and Gregory Berkolaiko and Alexandra Rodkina}, journal={Stochastics}, year={2009}, volume={81}, pages={127 - 99} }

We consider the stochastic difference equation where f and g are nonlinear, bounded functions, is a sequence of independent random variables, and h>0 is a nonrandom parameter. We establish results on asymptotic stability and instability of the trivial solution . We also show that, for some natural choices of f and g, the rate of decay of is approximately polynomial: there exists such that decays faster than but slower than , for any . It turns out that, if decays faster than as , the polynomial…

## 44 Citations

### On the local dynamics of polynomial difference equations with fading stochastic perturbations

- Mathematics
- 2010

We examine the stability-instability behaviour of a polynomial difference equa- tion with state-independent, asymptotically fading stochastic perturbations. We find that the set of initial values can…

### Global stabilization and destabilization by the state dependent noise with particular distributions

- MathematicsPhysica D: Nonlinear Phenomena
- 2020

### Asymptotic Stability of a Jump-Diffusion Equation and Its Numerical Approximation

- MathematicsSIAM J. Sci. Comput.
- 2008

A rigorous verification that both linear stability and instability are reproduced for small step sizes is given, which is known not to hold for general, nonlinear problems.

### On convergence of solutions to difference equations with additive perturbations

- Mathematics
- 2016

Various types of stabilizing controls lead to a deterministic difference equation with the following property: once the initial value is positive, the solution tends to the unique positive…

### Discretized It^ o Formula and Stability of Stochastic Dierence equations

- Mathematics
- 2009

We discuss a new variant of Discretized Ito formula originally developed in [1]. This variant significantly relaxes the assumption on the rate of decay of the tails of the noise’s density, presented…

### Strong convergence and stability of backward Euler–Maruyama scheme for highly nonlinear hybrid stochastic differential delay equation

- Mathematics
- 2015

In the paper, our main aim is to investigate the strong convergence and almost surely exponential stability of an implicit numerical approximation under one-sided Lipschitz condition and polynomial…

### Explosions and unbounded growth in nonlinear delay differential equations: Numerical and asymptotic analysis

- Mathematics
- 2011

This thesis investigates the asymptotic behaviour of a scalar, nonlinear dierential equation with a fixed delay, and examines whether the properties of this equation can be
replicated by an…

### Asymptotic moment boundedness of the numerical solutions of stochastic differential equations

- MathematicsJ. Comput. Appl. Math.
- 2013

### Instability and stability of solutions of systems of nonlinear stochastic difference equations with diagonal noise

- Mathematics
- 2014

We prove results about almost sure instability and stability of the equilibrium of a system of nonlinear stochastic difference equation with a small parameter h. The structure of the system is…

## References

SHOWING 1-10 OF 35 REFERENCES

### Almost sure polynomial asymptotic stability of stochastic difference equations

- Mathematics
- 2008

In this paper, we establish the almost sure asymptotic stability and decay results for solutions of an autonomous scalar difference equation with a nonhyperbolic equilibrium at the origin, which is…

### Rates of decay and growth of solutions to linear stochastic differential equations with state-independent perturbations

- Mathematics
- 2005

The paper studies the almost sure asymptotic convergence to zero of solutions of perturbed linear stochastic differential equations, where the unperturbed equation has an equilibrium at zero, and all…

### Almost sure convergence of solutions to non-homogeneous stochastic difference equation

- Mathematics
- 2006

We consider a non-homogeneous non-linear stochastic difference equation and its linear counterpart both with initial value , non-random decaying free coefficient S n and independent random variables…

### On boundedness and stability of solutions of nonlinear difference equation with nonmartingale type noise

- Mathematics
- 2001

Consider a stochastic difference equation with the Volterra type nonlinear main term G and Volterra type noise Functions Gf, σ supposed to be random, x;i is a martingale—difference. In general so Eq,…

### On Asymptotic Behavior of Solutions to Linear Discrete Stochastic Equation

- Mathematics
- 2004

We consider stochastic linear difference equations Xn+1 = Xn 3 1 + ξn+1 + Sn, n = 1, 2, . . . , X0 = x0, where ξi are i.i.d. random variables with Eξi = 0 and Var ln |1 + ξi| <∞. In the homogeneous…

### Almost Sure and Moment Exponential Stability in the Numerical Simulation of Stochastic Differential Equations

- MathematicsSIAM J. Numer. Anal.
- 2007

This analysis is motivated by an example of an exponentially almost surely stable nonlinear SDE for which the Euler-Maruyama method fails to reproduce this behavior for any nonzero timestep, and shows that when the SDE obeys a linear growth condition, EM recovers almost surely exponential stability very well.

### On stochastic stabilization of difference equations

- Mathematics
- 2006

We consider unstable scalar deterministic difference equation
$x_{n+1}=x_n(1+a_nf(x_n))$, $n\ge 1$, $x_0=a$.
We show how this equation can be stabilized by adding the random
noise term…

### Strong Convergence of Euler-Type Methods for Nonlinear Stochastic Differential Equations

- MathematicsSIAM J. Numer. Anal.
- 2002

This work gives a convergence result for Euler--Maruyama requiring only that the SDE is locally Lipschitz and that the pth moments of the exact and numerical solution are bounded for some p >2 and shows that the optimal rate of convergence can be recovered if the drift coefficient is also assumed to behave like a polynomial.

### Global asymptotic stability of solutions of cubic stochastic difference equations

- Mathematics
- 2004

Global almost sure asymptotic stability of solutions of some nonlinear stochastic difference equations with cubic-type main part in their drift and diffusive part driven by square-integrable…

### Pathwise non-exponential decay rates of solutions of scalarnonlinear stochastic differential equations

- Mathematics
- 2006

This paper studies the pathwise asymptotic stability of the zero
solution of scalar stochastic differential equation of Ito type.
Specifically, we provide conditions for solutions to converge to …