Numerical approximation of SDEs with fractional noise and distributional drift

TL;DR

This paper studies numerical approximation of SDEs with fractional noise and distributional drift, showing strong well-posedness and convergence of a tamed Euler scheme in the Catellier–Gubinelli regime $\gamma>1-\frac{1}{2H}$. It proves an explicit strong convergence rate in the subcritical case and non-explicit rates at the threshold and in the limit case, using new regularisation properties of discrete-time fBm and a critical Grönwall-type lemma. The main technical machinery relies on Besov-space regularity of the drift, stochastic sewing, and careful analysis of the discrete-time fractional noise, avoiding Girsanov transforms to control growth of the drift. As a byproduct, strong existence and pathwise uniqueness are recovered for these irregular SDEs, and the theory is complemented by concrete examples and simulations illustrating the rates. The results unify and extend prior works on rough drift SDEs driven by Brownian or fractional noise and provide practical numerical schemes for simulating such systems.

Abstract

We study the numerical approximation of SDEs with singular drifts (including distributions) driven by a fractional Brownian motion. Under the Catellier-Gubinelli condition that imposes the regularity of the drift to be strictly greater than $1-1/(2H)$, we obtain an explicit rate of convergence of a tamed Euler scheme towards the SDE, extending results for bounded drifts. Beyond this regime, when the regularity of the drift is $1-1/(2H)$, we derive a non-explicit rate. As a byproduct, strong well-posedness for these equations is recovered. Proofs use new regularising properties of discrete-time fBm and a new critical Grönwall-type lemma. We present examples and simulations.

Figures (3)

• Figure 1: The hashed region represents admissible values of $\gamma$ and $H$ under \ref{['eqcond-gamma-p-H']}, when $\gamma <0$. On the blue lines, we read the order of convergence: for a fixed point $(\gamma, H)$ with $\gamma<0$ (see the red cross), one can read the order of convergence given $\phi(\gamma) \colon =\frac{1}{2(1-\gamma)}$ by projecting the point on $\phi$ from below (see the blue arrow). For $\gamma>0$, the order varies with $H$, and therefore several curves are used to represent it.
• Figure 2: Plot of the logarithm of the strong error ($y$-axis) against $h$ ($x$-axis) for a bounded drift. Left: Equation \ref{['eqsimubounded']} ($d=1$), the calculated slopes of the linear regression are approximately $0.51, 0.53, 0.53$. - Right: Equation \ref{['eqind2D']}$(d=2)$, the calculated slopes are approximately $0.53, 0.51, 0.50$. The standard deviation are plotted in dashed lines. For different values of $H<1/2$, and in both dimension $1$ and $2$, we observe that the numerical order of convergence is close to the theoretical order $1/2$.
• Figure 3: Plot of the logarithm of the strong error ($y$-axis) against $h$ ($x$-axis) for a Dirac drift in dimension $d=1$, namely Equation \ref{['eqskewfbm']}. For several values of $H<1/2$, the calculated slopes of the linear regression are, from top to bottom, approximately $0.30, 0.29, 0.23$. The standard deviations are plotted in dashed lines. We observe that the numerical order of convergence is a bit far from the theoretical order $1/4$, hence further investigation (considering a larger number of realisations or smaller time-steps) is required to achieve a better rate. In particular, the difficulty comes from the strong singularity of the Dirac drift and a less precise simulation of the fBm as $H$ gets smaller, which could explain why the rate is further from $1/4$ when $H$ is smaller.

Theorems & Definitions (49)

• Definition 2.1
• Definition 2.2
• Theorem 2.3: anzeletti2021regularisationGaleatiGerencser
• Remark 2.4
• Theorem 2.5
• Remark 2.6
• Corollary 2.7
• Remark 2.8
• Corollary 2.9
• Remark 2.10