Weak Convergence Of Tamed Exponential Integrators for Stochastic Differential Equations

TL;DR

The paper addresses weak convergence of taming exponential integrators for SDEs with non-globally Lipschitz drift by leveraging the linear GBM flow through a propagator $\mathbf{\Phi}_{t,t_0}$ and a taming term $F^{tm}_{\Delta t}$. It proves uniform moment bounds and a first-order weak convergence rate (for $\phi\in C^4_b$) using the Kolmogorov equation, and provides a detailed MLMC-based numerical comparison between GBM-based taming and standard exponential taming, highlighting regimes where each method performs best. The contributions include (i) a rigorous order-one weak convergence result under precise commutativity and smoothness assumptions, (ii) a practical assessment showing GBM-based taming can avoid restrictive time-steps for linear diffusion, and (iii) extensions to nonlinear diffusion requiring step-size restrictions. These results offer guidance for efficiently simulating stiff SDEs with multiplicative noise while providing provable accuracy guarantees.

Abstract

We prove weak convergence of order one for a class of exponential based integrators for SDEs with non-globally Lipschtiz drift. Our analysis covers tamed versions of Geometric Brownian Motion (GBM) based methods as well as the standard exponential schemes. The numerical performance of both the GBM and exponential tamed methods through four different multi-level Monte Carlo techniques are compared. We observe that for linear noise the standard exponential tamed method requires severe restrictions on the stepsize unlike the GBM tamed method.

Figures (4)

• Figure 1: Comparison of different approaches to estimate the weak error for $d=1$ with $\beta_1=0.1$ and $\beta_2=0$.
• Figure 2: Comparison of 3 different approaches to estimate the weak error for $d=4$ with $\beta_1=0.1$ and $\beta_2=0$. In (d) we see that the exponential tamed method is not well behaved for large time step size.
• Figure 3: Convergence for $d=50$ and $d=100$ with $\beta_1=0.1$ and $\beta_2=0$. For larger time step sizes the exponential tamed method has a large variance and we do not see convergence until smaller time step sizes.
• Figure 4: Weak convergence for $d=4$ with $\beta_1=\beta_2=0.1$. We observe rate one convergence. A time step size restriction is required.

Theorems & Definitions (21)

• Theorem 1: cerrai,Theorem 1.3.5
• Theorem 2: cerrai, Theorem 1.3.6
• Theorem 3: cerrai, Theorem 1.6.2
• Remark 1
• Lemma 1
• proof
• Remark 2
• Theorem 4
• Theorem 5
• Lemma 2