A Tail-Respecting Explicit Numerical Scheme for Lévy-Driven SDEs With Superlinear Drifts
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912830485168128 |
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| author | Aryasova, Olga Kulyk, Oleksii Pavlyukevich, Ilya |
| author_facet | Aryasova, Olga Kulyk, Oleksii Pavlyukevich, Ilya |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_07255 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Tail-Respecting Explicit Numerical Scheme for Lévy-Driven SDEs With Superlinear Drifts Aryasova, Olga Kulyk, Oleksii Pavlyukevich, Ilya Probability 60H35, 60G51, 60H10, 65C05, 65C30 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. |
| title | A Tail-Respecting Explicit Numerical Scheme for Lévy-Driven SDEs With Superlinear Drifts |
| topic | Probability 60H35, 60G51, 60H10, 65C05, 65C30 |
| url | https://arxiv.org/abs/2504.07255 |