A Tail-Respecting Explicit Numerical Scheme for Lévy-Driven SDEs With Superlinear Drifts
Olga Aryasova, Oleksii Kulyk, Ilya Pavlyukevich
TL;DR
The paper develops a tail-respecting explicit Lie-Trotter–Euler scheme for multivariate SDEs with a superlinear κ-dissipative drift driven by a Lévy process with finite p-th moment. It proves strong L^q convergence rates and uniform strong convergence with rates γ and δ that depend on the drift growth, Lipschitz constants, and the Lévy-noise moment parameter p, while preserving the solution’s moment structure in both dissipative and Brownian-only settings. The analysis leverages a two-step splitting via the ODE flow Φ for the dissipative part and an Euler-type update for the remainder, together with semimartingale techniques and Lyapunov methods to establish moment bounds and tail behavior. Numerical simulations illustrate the limitations of naive Euler schemes (e.g., NaNs) and demonstrate the moment-preserving properties and convergence of the proposed splitting method, including in heavy-tailed jump scenarios. The results provide practically relevant guidance for simulating complex dissipative systems with heavy-tailed noise, offering explicit rates and rigorous guarantees for the proposed explicit scheme.
Abstract
We present an explicit numerical approximation scheme, denoted by $\{X^n\}$, for the effective simulation of solutions $X$ to a multivariate stochastic differential equation (SDE) with a superlinearly growing $κ$-dissipative drift, where $κ>1$, driven by a multiplicative heavy-tailed Lévy process that has a finite $p$-th moment, with $p>0$. We show that the strong $L^q$-convergence $\sup_{t\in[0,T]}\mathbf E \|X^n_t-X_t\|^q=\mathcal O (h_n^γ)$ holds true for any $q\in (0,p+κ-1)$, which is exactly the range where the $q$-moment of the solution is known to be finite. Additionally, for any $q\in (0,p)$ we establish strong uniform convergence: $\mathbf E\sup_{t\in[0,T]} \|X^n_t-X_t\|^q=\mathcal{O} ( h_n^δ )$. In both cases we determine the convergence rates $γ$ and $δ$. In the special case of SDEs driven solely by a Brownian motion, our numerical scheme preserves super-exponential moments of the solution. The scheme $\{X^n\}$ is realized as a combination of a well-known Euler method with a Lie-Trotter type splitting technique.