Approximation of SDEs -- a stochastic sewing approach

Abstract

We give a new take on the error analysis of approximations of stochastic differential equations (SDEs), utilizing and developing the stochastic sewing lemma of Lê (2020). 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 $H\in (0,1)$ and the drift is ${\mathcal{C}}^{\alpha }$, $\alpha \in [0,1]$ and $\alpha >1-1/(2H)$, we show the strong $L_p$ and almost sure rates of convergence to be $((1/2+\alpha H)\wedge 1)-\epsilon$, 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, Gubinelli (2016). 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 ${\mathcal{C}}^{\alpha }$ drift, for any $\epsilon ,\alpha >0$.

51 pages. Accepted version

References in corpus (3)

• On the regularisation of the noise for the Euler-Maruyama scheme with irregular drift
• The Euler-Maruyama Scheme for SDEs with Irregular Drift: Convergence Rates via Reduction to a Quadrature Problem
• A numerical scheme for stochastic differential equations with distributional drift

Cited by in corpus (7)

• Quantifying a convergence theorem of Gyöngy and Krylov
• A numerical scheme for stochastic differential equations with distributional drift
• Regularisation by fractional noise for one-dimensional differential equations with distributional drift
• Optimal rate of convergence for approximations of SPDEs with non-regular drift
• Regularisation by regular noise
• Solution theory of fractional SDEs in complete subcritical regimes
• Convergence rate of numerical scheme for SDEs with a distributional drift in Besov space