Approximation of SDEs: a stochastic sewing approach

We give a new take on the error analysis of approximations of stochastic differential equations (SDEs), utilizing and developing the stochastic sewing lemma of Le (Electron J Probab 25:55, 2020. https://doi.org/10.1214/20-EJP442). This approach allows one to exploit regularization 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 is an element of (0, 1) and the drift is C-alpha, alpha is an element of [0, 1] and alpha > 1 - 1/(2H), we show the strong L-p and almost sure rates of convergence to be ((1/2+alpha H) boolean AND 1) - epsilon, for any epsilon > 0. Our conditions on the regularity of the drift are optimal in the sense that they coincide with the conditions needed for the strong uniqueness of solutions from Catellier and Gubinelli (Stoch Process Appl 126(8):2323-2366, 2016. https://doi.org/10.1016/j.spa.2016.02.002). In a second application we consider the approximation of SDEs driven by multiplicative standard Brownian noise where we derive the almost optimal rate of convergence 1/2 - epsilon of the Euler-Maruyama scheme for C-alpha drift, for any epsilon, alpha > 0.