A martingale approach to noncommutative stochastic calculus
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| author | Jekel, David A. Kemp, Todd A. Nikitopoulos, Evangelos A. |
| author_facet | Jekel, David A. Kemp, Todd A. Nikitopoulos, Evangelos A. |
| contents | We present a new approach to noncommutative stochastic calculus that is, like the classical theory, based primarily on the martingale property. Using this approach, we introduce a general theory of stochastic integration and quadratic (co)variation for a certain class of noncommutative processes, analogous to semimartingales, that includes both the $q$-Brownian motions and classical matrix-valued Brownian motions. As applications, we obtain Burkholder--Davis--Gundy inequalities (with $p \geq 2$) for continuous-time noncommutative martingales and a noncommutative Itô's formula for "adapted $C^2$ maps," including trace $\ast$-polynomial maps and operator functions associated to the noncommutative $C^2$ scalar functions $\mathbb{R} \to \mathbb{C}$ introduced by Nikitopoulos, as well as the more general multivariate tracial noncommutative $C^2$ functions introduced by Jekel, Li, and Shlyakhtenko. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_09856 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A martingale approach to noncommutative stochastic calculus Jekel, David A. Kemp, Todd A. Nikitopoulos, Evangelos A. Operator Algebras Probability 46L54, 60H05 (Primary) 46L52 (Secondary) We present a new approach to noncommutative stochastic calculus that is, like the classical theory, based primarily on the martingale property. Using this approach, we introduce a general theory of stochastic integration and quadratic (co)variation for a certain class of noncommutative processes, analogous to semimartingales, that includes both the $q$-Brownian motions and classical matrix-valued Brownian motions. As applications, we obtain Burkholder--Davis--Gundy inequalities (with $p \geq 2$) for continuous-time noncommutative martingales and a noncommutative Itô's formula for "adapted $C^2$ maps," including trace $\ast$-polynomial maps and operator functions associated to the noncommutative $C^2$ scalar functions $\mathbb{R} \to \mathbb{C}$ introduced by Nikitopoulos, as well as the more general multivariate tracial noncommutative $C^2$ functions introduced by Jekel, Li, and Shlyakhtenko. |
| title | A martingale approach to noncommutative stochastic calculus |
| topic | Operator Algebras Probability 46L54, 60H05 (Primary) 46L52 (Secondary) |
| url | https://arxiv.org/abs/2308.09856 |