Approximation of SDEs -- a stochastic sewing approach

Oleg Butkovsky, Konstantinos Dareiotis, Máté Gerencsér

Published 2019-09-17Version 1

We give a new take on the error analysis of approximations of stochastic differential equations (SDEs), utilising the stochastic sewing lemma [L\^e '18]. This approach allows one to exploit regularisation by noise effects in obtaining convergence rates. In our first application we show convergence (to our knowledge for the first time) of the Euler-Maruyama scheme for SDEs driven by fractional Brownian motions with non-regular drift. When the Hurst parameter is $H\in(0,1)$ and the drift is $\mathcal{C}^\alpha$, $\alpha>2-1/H$, we show the strong $L_p$ and almost sure rates of convergence to be $1/2+\alpha(1/2\wedge H)-\varepsilon$, for any $\varepsilon>0$. As another application we consider the approximation of SDEs driven by multiplicative standard Brownian noise where we derive the almost optimal rate of convergence $1/2-\varepsilon$ of the Euler-Maruyama scheme for $\mathcal{C}^\alpha$ drift, for any $\varepsilon,\alpha>0$.