A Tail-Respecting Splitting Numerical Scheme for Lévy-Driven SDEs With Superlinear Drifts

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 $\hat{I}º$-dissipative drift, where $\hat{I}º>1$, 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 ${\sup }_{t\in [0,T]}\mathbf{E}\Vert X_t^n-X_t{\Vert }^{p_X}=\mathcal{O}(h_n^3)$ holds for any $p_X\in (0,p+\hat{I}º-1)$, 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: $\mathbf{E}{\sup }_{t\in [0,T]}\Vert X_t^n-X_t{\Vert }^{p_X}=\mathcal{O}(h_n^{\hat{I}}´)$. In both cases we determine the convergence rates ${\hat{I}}^3$ and $\hat{I}´$. 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.

43 pages