On the $\varepsilon$-Euler-Maruyama scheme for time-inhomogeneous jump-driven SDEs

TL;DR

This paper introduces an advanced Euler-type scheme for simulating time-inhomogeneous jump-driven SDEs, achieving optimal strong and weak convergence rates, and extends existing methods to handle complex jump structures with practical applications.

Contribution

It proposes a novel two-parameters Euler scheme for jump SDEs with proven optimal convergence rates, extending the Asmussen-Rosiński approach to complex jump structures and providing comprehensive theoretical and numerical validation.

Findings

Achieves strong convergence rate of order 1/p for the scheme.

Obtains weak convergence rate of 1/n + ε^{3-β} under regularity conditions.

Numerical experiments confirm the theoretical convergence rates.

Abstract

We consider a class of general SDEs with a jump integral term driven by a time-inhomogeneous Poisson random measure. We propose a two-parameters Euler-type scheme for this SDE class and prove an optimal rate for the strong convergence with respect to the $L^p(\Omega)$-norm and for the weak convergence, considering integration over $n$ uniform time-steps. One of the primary issues to address in this context is the approximation of the noise structure when it can no longer be expressed as the increment of random variables. We extend the Asmussen-Rosi\'nski approach to the case of a fully dependent jump coefficient and time-dependent Poisson compensation, handling contribution of jumps smaller than $\varepsilon$ with an appropriate Gaussian substitute and exact simulation for the large jumps contribution. For any $p \geq 2$, under hypotheses required to control the $L^p$-moments of the process, we obtain a strong convergence rate of order $1/p$. Under standard regularity hypotheses on the coefficients, we obtain a weak convergence rate of $1/n+\varepsilon^{3-\beta}$, where $\beta$ is the Blumenthal-Getoor index of the underlying L\'evy measure. We compare this scheme with the Rubenthaler's approach where the jumps smaller than $\varepsilon$ are neglected, providing strong and weak rates of convergence in that case too. The theoretical rates are confirmed by numerical experiments afterwards. We apply this model class for some anomalous diffusion model related to the dynamics of rigid fibres in turbulence.