# On the weak convergence rate of an exponential Euler scheme for SDEs governed by coefficients with superlinear growth

## TL;DR

This paper analyzes the weak convergence rate of an exponential Euler scheme for one-dimensional SDEs with superlinear coefficients, establishing a convergence rate of order one under certain conditions and supporting findings with numerical experiments.

## Contribution

It introduces a semi-explicit exponential Euler scheme for SDEs with superlinear growth and proves its weak convergence rate of order one, extending existing methods.

## Key findings

- Weak convergence rate of order one established
- Scheme effective for SDEs with superlinear coefficients
- Numerical experiments support theoretical results

## Abstract

We consider the problem of the approximation of the solution of a one-dimensional SDE with non-globally Lipschitz drift and diffusion coefficients behaving as $x^\alpha$, with $\alpha>1$. We propose an (semi-explicit) exponential-Euler scheme and study its convergence through its weak approximation error. To this aim, we analyze the $C^{1,4}$ regularity of the solution of the associated backward Kolmogorov PDE using its Feynman-Kac representation and the flow derivative of the involved processes. From this, under some suitable hypotheses on the parameters of the model ensuring the control of its positive moments, we recover a rate of weak convergence of order one for the proposed exponential Euler scheme. Finally, numerical experiments are shown in order to support and complement our theoretical result.