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 $\kappa$-dissipative drift, where $\kappa>1$, driven by a multiplicative heavy-tailed Lévy process that has a finite $p$-th moment, with $p>0$. We show that the strong $L^{p_X}$-convergence $\sup_{t\in[0,T]}\mathbf E \|X^n_t-X_t\|^{p_X}=\mathcal O (h_n^{\gamma})$ holds for any $p_X\in (0,p+\kappa-1)$, which is exactly the range where the $p_X$-moment of the solution is known to be finite. Additionally, for any $p_X\in (0,p)$ we establish strong uniform convergence: $\mathbf E\sup_{t\in[0,T]} \|X^n_t-X_t\|^{p_X}=\mathcal{O} ( h_n^{\delta} )$. In both cases we determine the convergence rates $\gamma$ and $\delta$. 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.
From: Ilya Pavlyukevich [view email]
[v1]
Wed, 9 Apr 2025 20:16:46 UTC (36 KB)
[v2]
Mon, 19 Jan 2026 11:31:54 UTC (39 KB)
[v3]
Mon, 15 Jun 2026 07:17:47 UTC (116 KB)
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