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**1 - 9**of**9**### Convergence of numerical methods for stochastic differential equations in mathematical finance

, 1204

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### 2 Semi-Parametric Drift and Diffusion Estimation for Multiscale Diffusions

, 2012

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### Improving MLMC for SDEs with application to the Langevin

, 2014

"... This paper applies several well-known tricks from deterministic differential equations to improve the efficiency of the Multilevel Monte Carlo (MLMC) method for stochastic differential equations (SDEs) and especially the Langevin equation. We use modified equations analysis to circumvent the need fo ..."

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This paper applies several well-known tricks from deterministic differential equations to improve the efficiency of the Multilevel Monte Carlo (MLMC) method for stochastic differential equations (SDEs) and especially the Langevin equation. We use modified equations analysis to circumvent the need for a strong-approximation theory for the integrator. This gives greater freedom in the application of MLMC. For example, this allows us to replace Gaussian increments with discrete approximations that satisfy certain moment conditions and that allow an exact integration on the coarser levels. To be able to use large time steps to exploit this fully, we employ stable geometric integrators based on operator splitting for Langevin-type equations. The final enhancement we investigate is extrapolation, which is a well-known technique to reduce the bias and combines in an very natural way with MLMC. Combined, our enhancements lead to an order of magnitude improvement in efficiency, as we show in numerical experiments for MLMC applied to the Langevin equation. Keywords numerical solution of stochastic differential equations, modified equations,

### 4 CONVERGENCE OF TAMED EULER SCHEMES FOR A CLASS OF STOCHASTIC EVOLUTION EQUATIONS

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### Edinburgh Research Explorer

"... Convergence of tamed Euler schemes for a class of stochastic evolution equations Citation for published version: ..."

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Convergence of tamed Euler schemes for a class of stochastic evolution equations Citation for published version: