# Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients

@article{Hutzenthaler2012StrongCO, title={Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients}, author={Martin Hutzenthaler and Arnulf Jentzen and Peter E. Kloeden}, journal={Annals of Applied Probability}, year={2012}, volume={22}, pages={1611-1641} }

On the one hand, the explicit Euler scheme fails to converge strongly to the exact solution of a stochastic differential equation (SDE) with a superlinearly growing and globally one-sided Lipschitz continuous drift coefficient. On the other hand, the implicit Euler scheme is known to converge strongly to the exact solution of such an SDE. Implementations of the implicit Euler scheme, however, require additional computational effort. In this article we therefore propose an explicit and easily…

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## References

SHOWING 1-10 OF 44 REFERENCES

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.

Strong and weak divergence in finite time of Euler's method for stochastic differential equations with non-globally Lipschitz continuous coefficients

- MathematicsProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
- 2010

The stochastic Euler scheme is known to converge to the exact solution of a stochastic differential equation (SDE) with globally Lipschitz continuous drift and diffusion coefficients. Recent results…

Semi-Implicit Euler-Maruyama Scheme for Stiff Stochastic Equations

- Mathematics
- 1996

We discuss a semi-implicit time discretization scheme to approximate the solution of a kind of stiff stochastic differential equations. Roughly speaking, by stiffness for an SDE we mean that the…

Convergence of the Stochastic Euler Scheme for Locally Lipschitz Coefficients

- MathematicsFound. Comput. Math.
- 2011

This paper establishes convergence of the Monte Carlo Euler method for a large class of one-dimensional stochastic differential equations whose drift functions have at most polynomial growth.

A Note on the Rate of Convergence of the Euler–Maruyama Method for Stochastic Differential Equations

- Mathematics
- 2008

Abstract The recent article [2] reveals the strong convergence of the Euler–Maruyama solution to the exact solution of a stochastic differential equation under the local Lipschitz condition. However,…

Strong convergence rates for backward Euler–Maruyama method for non-linear dissipative-type stochastic differential equations with super-linear diffusion coefficients

- Mathematics
- 2013

In this work, we generalize the current theory of strong convergence rates for the backward Euler–Maruyama scheme for highly non-linear stochastic differential equations, which appear in both…

Stochastic Hamiltonian Systems : Exponential Convergence to the Invariant Measure , and Discretization by the Implicit Euler Scheme

- Mathematics
- 2002

In this paper we carefully study the large time behaviour of u(t, x, y) := Ex,y f(Xt, Yt)− ∫ f dμ, where (Xt, Yt) is the solution of a stochastic Hamiltonian dissipative system with non gbally…

Effectiveness of implicit methods for stiff stochastic differential equations

- Computer Science
- 2008

It is shown that implicit methods in general fail to capture the effective dynamics at the slow time scale due to the fact that such implicit methods cannot correctly capture non-Dirac invariant distributions when the time step size is much larger than the relaxation time of the system.

Pathwise accuracy and ergodicity of metropolized integrators for SDEs

- Mathematics
- 2009

Metropolized integrators for ergodic stochastic differential equations (SDEs) are proposed that (1) are ergodic with respect to the (known) equilibrium distribution of the SDEs and (2) approximate…

The numerical solution of stochastic differential equations

- MathematicsThe Journal of the Australian Mathematical Society. Series B. Applied Mathematics
- 1977

Abstract A method is proposed for the numerical solution of Itô stochastic differential equations by means of a second-order Runge–Kutta iterative scheme rather than the less efficient Euler…