Butkovsky Oleg, Dareiotis Konstantinos, Gerencsér Máté
Weierstrass Institute, Mohrenstraße 39, 10117 Berlin, Germany.
University of Leeds, Woodhouse, Leeds, LS2 9JT UK.
Probab Theory Relat Fields. 2021;181(4):975-1034. doi: 10.1007/s00440-021-01080-2. Epub 2021 Jul 30.
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 (Electron J Probab 25:55, 2020. 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 and the drift is , and , we show the strong and almost sure rates of convergence to be , for any . 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. 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 of the Euler-Maruyama scheme for drift, for any .
我们利用并发展了勒（Lê）的随机缝合引理（《电子概率杂志》25:55，2020. 10.1214/20 - EJP442），对随机微分方程（SDEs）近似的误差分析给出了一种新的视角。这种方法使人们能够在获得收敛速率时利用噪声效应进行正则化。在我们的第一个应用中，我们展示了（据我们所知首次）由具有非正则漂移的分数布朗运动驱动的SDEs的欧拉 - 丸山（Euler - Maruyama）格式的收敛性。当赫斯特参数为 且漂移为 、 和 时，我们表明对于任意 ，强收敛速率和几乎必然收敛速率均为 。我们对漂移正则性的条件在某种意义上是最优的，即它们与卡特利尔（Catellier）和古比内利（Gubinelli）（《随机过程及其应用》126(8):2323 - 2366，2016. 10.1016/j.spa.2016.02.002）中解的强唯一性所需的条件一致。在第二个应用中，我们考虑由乘性标准布朗噪声驱动的SDEs的近似，其中我们推导了对于任意 ，欧拉 - 丸山格式对于 漂移的几乎最优收敛速率 。
