Pathwise convergence of the Euler scheme for rough and stochastic differential equations
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866908538982367232 |
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| author | Allan, Andrew L. Kwossek, Anna P. Liu, Chong Prömel, David J. |
| author_facet | Allan, Andrew L. Kwossek, Anna P. Liu, Chong Prömel, David J. |
| contents | The convergence of the first order Euler scheme and an approximative variant thereof, along with convergence rates, are established for rough differential equations driven by càdlàg paths satisfying a suitable criterion, namely the so-called Property (RIE), along time discretizations with vanishing mesh size. This property is then verified for almost all sample paths of Brownian motion, Itô processes, Lévy processes and general càdlàg semimartingales, as well as the driving signals of both mixed and rough stochastic differential equations, relative to various time discretizations. Consequently, we obtain pathwise convergence in p-variation of the Euler--Maruyama scheme for stochastic differential equations driven by these processes. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2309_16489 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Pathwise convergence of the Euler scheme for rough and stochastic differential equations Allan, Andrew L. Kwossek, Anna P. Liu, Chong Prömel, David J. Probability Numerical Analysis 65C30, 65L20, 60H35, 60L20 The convergence of the first order Euler scheme and an approximative variant thereof, along with convergence rates, are established for rough differential equations driven by càdlàg paths satisfying a suitable criterion, namely the so-called Property (RIE), along time discretizations with vanishing mesh size. This property is then verified for almost all sample paths of Brownian motion, Itô processes, Lévy processes and general càdlàg semimartingales, as well as the driving signals of both mixed and rough stochastic differential equations, relative to various time discretizations. Consequently, we obtain pathwise convergence in p-variation of the Euler--Maruyama scheme for stochastic differential equations driven by these processes. |
| title | Pathwise convergence of the Euler scheme for rough and stochastic differential equations |
| topic | Probability Numerical Analysis 65C30, 65L20, 60H35, 60L20 |
| url | https://arxiv.org/abs/2309.16489 |