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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 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 H∈(0,1) and the drift is Cα , α∈[0,1] and α>1-1/(2H) , we show the strong Lp and almost sure rates of convergence to be ((1/2+αH)∧1)-ε , for any ε>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. 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-ε of the Euler-Maruyama scheme for Cα drift, for any ε,α>0 .
Singular paths spaces and applications
Motivated by recent applications in rough volatility and regularity structures, notably the notion of singular modeled distribution, we study paths, rough paths and related objects with a quantified singularity at zero. In a pure path setting, this allows us to leverage on existing SLE Besov estimates to see that SLE traces takes values in a singular Hölder space, which quantifies a well-known boundary effect in the regime κ<1. We then consider the integration theory against singular rough paths and some extensions thereof. This gives a method to reconcile, from a regularity structure point of view, different singular kernels used to construct (fractional) rough volatility models and an effective reduction to the stationary case which is crucial to apply general renormalization methods.