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 (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 Hin (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) -varepsilon , for any varepsilon >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-varepsilon of the Euler–Maruyama scheme for mathcal {C}^alpha drift, for any varepsilon ,alpha >0.
Highlights
Where B H is a d-dimensional fractional Brownian motion with Hurst parameter H ∈ (0, 1)
In our first application we show convergence of the Euler–Maruyama scheme for SDEs driven by fractional Brownian motions with nonregular drift
Austria where B H is a d-dimensional fractional Brownian motion with Hurst parameter H ∈ (0, 1). It is known [8, Theorem 1.9] that this equation has a unique strong solution if b belongs to the Hölder–Besov space Cα and α > 1 − 1/(2H )
Summary
We begin by introducing the basic notation. Consider a probability space ( , F, P). Carrying a d-dimensional two-sided Brownian motion (Wt )t∈R. For α ∈ (0, 1] and a function f : Q → V , where Q ⊂ Rk and (V , | · |) is a normed space, we set [. For α ∈ (0, ∞) we denote by Cα(Q, V ) the space of all functions f : Q → V having derivatives ∂ f for all multi-indices ∈ (Z+)k with | | < α such that f Cα(Q,V ) :=. We set C0(Q, V ) to be the space of bounded measurable functions with the supremum norm. Convention on constants: throughout the paper N denotes a positive constant whose value may change from line to line; its dependence is always specified in the corresponding statement
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