74 research outputs found
Convergence of the Euler-Maruyama method for multidimensional SDEs with discontinuous drift and degenerate diffusion coefficient
We prove strong convergence of order for arbitrarily small
of the Euler-Maruyama method for multidimensional stochastic
differential equations (SDEs) with discontinuous drift and degenerate diffusion
coefficient. The proof is based on estimating the difference between the
Euler-Maruyama scheme and another numerical method, which is constructed by
applying the Euler-Maruyama scheme to a transformation of the SDE we aim to
solve
Convergence of the tamed-Euler-Maruyama method for SDEs with discontinuous and polynomially growing drift
Numerical methods for SDEs with irregular coefficients are intensively
studied in the literature, with different types of irregularities usually being
attacked separately. In this paper we combine two different types of
irregularities: polynomially growing drift coefficients and discontinuous drift
coefficients. For SDEs that suffer from both irregularities we prove strong
convergence of order of the tamed-Euler-Maruyama scheme from
[Hutzenthaler, M., Jentzen, A., and Kloeden, P. E., The Annals of Applied
Probability, 22(4):1611-1641, 2012]
On the complexity of strong approximation of stochastic differential equations with a non-Lipschitz drift coefficient
We survey recent developments in the field of complexity of pathwise
approximation in -th mean of the solution of a stochastic differential
equation at the final time based on finitely many evaluations of the driving
Brownian motion. First, we briefly review the case of equations with globally
Lipschitz continuous coefficients, for which an error rate of at least in
terms of the number of evaluations of the driving Brownian motion is always
guaranteed by using the equidistant Euler-Maruyama scheme. Then we illustrate
that giving up the global Lipschitz continuity of the coefficients may lead to
a non-polynomial decay of the error for the Euler-Maruyama scheme or even to an
arbitrary slow decay of the smallest possible error that can be achieved on the
basis of finitely many evaluations of the driving Brownian motion. Finally, we
turn to recent positive results for equations with a drift coefficient that is
not globally Lipschitz continuous. Here we focus on scalar equations with a
Lipschitz continuous diffusion coefficient and a drift coefficient that
satisfies piecewise smoothness assumptions or has fractional Sobolev regularity
and we present corresponding complexity results
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