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 (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 $((1/2+\alpha H)\wedge 1) -\varepsilon$, 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, 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 $1/2-\varepsilon$ of the Euler-Maruyama scheme for $\mathcal{C}^\alpha$ drift, for any $\varepsilon,\alpha>0$.

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Approximation of SDEs -- a stochastic sewing approach.

The strong rate of convergence of the Euler-Maruyama scheme for nondegenerate SDEs with irregular drift coefficients is considered. In the case of $\alpha$-Holder drift in the recent literature the rate $\alpha/2$ was proved in many related situations. By exploiting the regularising effect of the noise more efficiently, we show that the rate is in fact arbitrarily close to $1/2$ for all $\alpha>0$. The result extends to Dini continuous coefficients, while in $d=1$ also to all bounded measurable coefficients.

On the regularisation of the noise for the Euler-Maruyama scheme with irregular drift

The goal of this paper is to approximate two kinds of McKean–Vlasov stochastic differential equations (SDEs) with irregular coefficients via weakly interacting particle systems. More precisely, propagation of chaos and convergence rate of Euler–Maruyama scheme associated with the consequent weakly interacting particle systems are investigated for McKean–Vlasov SDEs, where (1) the diffusion terms are Holder continuous by taking advantage of Yamada–Watanabe’s approximation approach and (2) the drifts are Holder continuous by freezing distributions followed by invoking Zvonkin’s transformation trick.

Approximations of McKean–Vlasov Stochastic Differential Equations with Irregular Coefficients

The explicit Euler scheme and similar explicit approximation schemes (such as the Milstein scheme) are known to diverge strongly and numerically weakly in the case of one-dimensional stochastic ordinary differential equations with superlinearly growing nonlinearities. It remained an open question whether such a divergence phenomenon also holds in the case of stochastic partial differential equations with superlinearly growing nonlinearities such as stochastic Allen-Cahn equations. In this work we solve this problem by proving that full-discrete exponential Euler and full-discrete linear-implicit Euler approximations diverge strongly and numerically weakly in the case of stochastic Allen-Cahn equations. This article also contains a short literature overview on existing numerical approximation results for stochastic differential equations with superlinearly growing nonlinearities.

Strong and weak divergence of exponential and linear-implicit Euler approximations for stochastic partial differential equations with superlinearly growing nonlinearities

In this paper, we are concerned with convergence rate of Euler–Maruyama scheme for stochastic differential equations with Holder–Dini continuous drifts. The key contributions are as follows: (i) by means of regularity of non-degenerate Kolmogrov equation, we investigate convergence rate of Euler–Maruyama scheme for a class of stochastic differential equations which allow the drifts to be Dini continuous and unbounded; (ii) by the aid of regularization properties of degenerate Kolmogrov equation, we discuss convergence rate of Euler–Maruyama scheme for a range of degenerate stochastic differential equations where the drifts are Holder–Dini continuous of order $$\frac{2}{3}$$
 with respect to the first component and are merely Dini-continuous concerning the second component.

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Convergence Rate of Euler–Maruyama Scheme for SDEs with Hölder–Dini Continuous Drifts

The existence and uniqueness of a mild solution for a class of functional stochastic partial differential equations with multiplicative noise and a locally Dini continuous drift are proved. In addition, under a reasonable condition the solution is non-explosive. Moreover, Harnack inequalities are derived for the associated semigroup under certain global conditions, which is new even in the case without delay.

Xing Huang

Papers

Convergence Rate of Euler–Maruyama Scheme for SDEs with Hölder–Dini Continuous Drifts

Mild Solutions and Harnack Inequality for Functional Stochastic Partial Differential Equations with Dini Drift