Approximation of SDEs - a stochastic sewing approach

We give a new take on the error analysis of approximations of stochastic differential equations (SDEs), utilising the stochastic sewing lemma [K. Le, ’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 $C^\alpha$, $\alpha>2-1/H$, we show the strong $L_p$ and almost sure rates of convergence to be $(1/2+\alpha H) \wedge 1-$. As another application we consider the approximation of SDEs driven by multiplicative standard Brownian noise where we derive the almost optimal rate of convergence $1/2-$ of the Euler-Maruyama scheme for $C^\alpha$ drift, for any $\alpha>0$.

This is a joint work with Oleg Butkovsky and Máté Gerencsér.