Cutoff stability of multivariate geometric Brownian motion
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2022
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| _version_ | 1866911026244485120 |
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| author | Barrera, G. Högele, M. A. Pardo, J. C. |
| author_facet | Barrera, G. Högele, M. A. Pardo, J. C. |
| contents | This article establishes cutoff convergence or abrupt convergence of three statistical quantities for multivariate (Hurwitz) stable geometric Brownian motion: the autocorrelation function, the Wasserstein distance between the current state and its degenerate limiting measure, and, finally, anti-concentration probabilities, which yield a fine-tuned trade-off between almost sure rates and the respective integrability of the random modulus of convergence using a quantitative Borel--Cantelli Lemma. We obtain in case of simultaneous diagonalizable drift and volatility matrices a complete representation of the mean square and derive nontrivial, sufficient and necessary mean square stability conditions, which include all real and imaginary parts of the volatility matrices' spectra. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2207_01666 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Cutoff stability of multivariate geometric Brownian motion Barrera, G. Högele, M. A. Pardo, J. C. Probability Mathematical Physics Dynamical Systems 37A25, 37A30, 34D20, 60H10, 37H15 This article establishes cutoff convergence or abrupt convergence of three statistical quantities for multivariate (Hurwitz) stable geometric Brownian motion: the autocorrelation function, the Wasserstein distance between the current state and its degenerate limiting measure, and, finally, anti-concentration probabilities, which yield a fine-tuned trade-off between almost sure rates and the respective integrability of the random modulus of convergence using a quantitative Borel--Cantelli Lemma. We obtain in case of simultaneous diagonalizable drift and volatility matrices a complete representation of the mean square and derive nontrivial, sufficient and necessary mean square stability conditions, which include all real and imaginary parts of the volatility matrices' spectra. |
| title | Cutoff stability of multivariate geometric Brownian motion |
| topic | Probability Mathematical Physics Dynamical Systems 37A25, 37A30, 34D20, 60H10, 37H15 |
| url | https://arxiv.org/abs/2207.01666 |