# On stochastic differential equations with arbitrary slow convergence rates for strong approximation

@article{Jentzen2015OnSD, title={On stochastic differential equations with arbitrary slow convergence rates for strong approximation}, author={Arnulf Jentzen and Thomas Muller-Gronbach and Larisa Yaroslavtseva}, journal={arXiv: Numerical Analysis}, year={2015} }

In the recent article [Hairer, M., Hutzenthaler, M., Jentzen, A., Loss of regularity for Kolmogorov equations, Ann. Probab. 43 (2015), no. 2, 468--527] it has been shown that there exist stochastic differential equations (SDEs) with infinitely often differentiable and globally bounded coefficients such that the Euler scheme converges to the solution in the strong sense but with no polynomial rate. Hairer et al.'s result naturally leads to the question whether this slow convergence phenomenon…

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## References

SHOWING 1-10 OF 34 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.

### Numerical Treatment of Stochastic Differential Equations

- Mathematics
- 1982

We define general Runge–Kutta approximations for the solution of stochastic differential equations (sde). These approximations are proved to converge in quadratic mean to the solution of an sde with…

### On a perturbation theory and on strong convergence rates for stochastic ordinary and partial differential equations with non-globally monotone coefficients

- Mathematics
- 2014

We develope a perturbation theory for stochastic differential equations (SDEs) by which we mean both stochastic ordinary differential equations (SODEs) and stochastic partial differential equations…

### Minimal Errors for Strong and Weak Approximation of Stochastic Differential Equations

- Mathematics, Computer Science
- 2008

A survey of results on minimal errors and optimality of algorithms for strong and weak approximation of systems of stochastic differential equations finds that the analysis of minimal errors leads to new algorithms that perform asymptotically optimal.

### Stochastic C-Stability and B-Consistency of Explicit and Implicit Euler-Type Schemes

- MathematicsJ. Sci. Comput.
- 2016

The convergence theorem is derived in a somewhat more abstract framework, but this paper also contains two more concrete examples of stochastically C-stable numerical one-step schemes: the split-step backward Euler method from Higham et al. (SIAM J Numer Anal 40(3):1041–1063, 2002) and a newly proposed explicit variant of the Euler-Maruyama scheme, the so called projected Euler– Maruyama method.

### The tamed Milstein method for commutative stochastic differential equations with non-globally Lipschitz continuous coefficients

- Mathematics
- 2011

For stochastic differential equations (SDEs) with a superlinearly growing and globally one-sided Lipschitz continuous drift coefficient, the classical explicit Euler scheme fails to converge strongly…

### AN AXIOMATIC APPROACH TO NUMERICAL APPROXIMATIONS OF STOCHASTIC PROCESSES

- Mathematics
- 2005

An axiomatic approach to the numerical approximation Y of some stochastic process X with values on a separable Hilbert space H is presented by means of Lyapunov-type control functions V . The…

### An Explicit Euler Scheme with Strong Rate of Convergence for Financial SDEs with Non-Lipschitz Coefficients

- MathematicsSIAM J. Financial Math.
- 2016

A modified explicit Euler-Maruyama discretisation scheme that allows us to prove strong convergence, with a rate, is presented, and under some regularity and integrability conditions, the optimal strong error rate is obtained.

### Loss of regularity for Kolmogorov equations

- Mathematics
- 2015

The celebrated Hormander condition is a sucient (and nearly necessary) condition for a second-order linear Kolmogorov partial dierential equation (PDE) with smooth coecients to be hypoelliptic. As a…

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

- Mathematics
- 2012

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…