2,322 research outputs found
Almost sure exponential stability of numerical solutions for stochastic delay differential equations
Using techniques based on the continuous and discrete semimartingale convergence theorems, this paper investigates if numerical methods may reproduce the almost sure exponential stability of the exact solutions to stochastic delay differential equations (SDDEs). The important feature of this technique is that it enables us to study the almost sure exponential stability of numerical solutions of SDDEs directly. This is significantly different from most traditional methods by which the almost sure exponential stability is derived from the moment stability by the Chebyshev inequality and the BorelāCantelli lemma
Almost sure exponential stability of the EulerāMaruyama approximations for stochastic functional differential equations
By the continuous and discrete nonnegative semimartingale convergence theorems, this paper investigates conditions under which the EulerāMaruyama (EM) approximations of stochastic functional differential equations (SFDEs) can share the almost sure exponential stability of the exact solution. Moreover, for sufficiently small stepsize, the decay rate as measured by the Lyapunov exponent can be reproduced arbitrarily accurately
Mean-square convergence and stability of the backward Euler method for stochastic differential delay equations with highly nonlinear growing coefficients
Over the last few decades, the numerical methods for stochastic differential
delay equations (SDDEs) have been investigated and developed by many scholars.
Nevertheless, there is still little work to be completed. By virtue of the
novel technique, this paper focuses on the mean-square convergence and
stability of the backward Euler method (BEM) for SDDEs whose drift and
diffusion coefficients can both grow polynomially. The upper mean-square error
bounds of BEM are obtained. Then the convergence rate, which is one-half, is
revealed without using the moment boundedness of numerical solutions.
Furthermore, under fairly general conditions, the novel technique is applied to
prove that the BEM can inherit the exponential mean-square stability with a
simple proof. At last, two numerical experiments are implemented to illustrate
the reliability of the theories
Almost sure and moment exponential stability in the numerical simulation of stochastic differential equations
Relatively little is known about the ability of numerical methods for stochastic differential equations (SDEs) to reproduce almost sure and small-moment stability. Here, we focus on these stability properties in the limit as the timestep tends to zero. Our analysis is motivated by an example of an exponentially almost surely stable nonlinear SDE for which the Euler-Maruyama (EM)method fails to reproduce this behavior for any nonzero timestep. We begin by showing that EM correctly reproduces almost sure and small-moment exponential stability for sufficiently small timesteps on scalar linear SDEs. We then generalize our results to multidimensional nonlinear SDEs. We show that when the SDE obeys a linear growth condition, EM recovers almost surely exponential stability very well. Under the less restrictive condition that the drift coefficient of the SDE obeys a one-sided Lipschitz condition, where EM may break down, we show that the backward Euler method maintains almost surely exponential stability
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