A criterion for absolute continuity relative to the law of fractional Brownian motion
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866916489545646080 |
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| author | Han, Xiyue Schied, Alexander |
| author_facet | Han, Xiyue Schied, Alexander |
| contents | Let $X$ be the sum of a fractional Brownian motion with Hurst parameter $H$ and an absolutely continuous and adapted drift process. We establish a simple criterion that guarantees that the law of $X$ is absolutely continuous with respect to the law of the original fractional Brownian motion. For $H<1/2$, the trajectories of the derivative of the drift need to be bounded by an almost surely finite random variable; for $H>1/2$, they need to satisfy a Hölder condition with some exponent larger than $2H-1$. These are almost-sure conditions, and no expectation requirements are imposed. For the case in which $X$ arises as the solution of a nonlinear stochastic integral equation driven by fractional Brownian motion, we provide a simple criterion on the drift coefficient under which the law of $X$ is automatically equivalent to the one of fractional Brownian motion. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_11824 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A criterion for absolute continuity relative to the law of fractional Brownian motion Han, Xiyue Schied, Alexander Probability Let $X$ be the sum of a fractional Brownian motion with Hurst parameter $H$ and an absolutely continuous and adapted drift process. We establish a simple criterion that guarantees that the law of $X$ is absolutely continuous with respect to the law of the original fractional Brownian motion. For $H<1/2$, the trajectories of the derivative of the drift need to be bounded by an almost surely finite random variable; for $H>1/2$, they need to satisfy a Hölder condition with some exponent larger than $2H-1$. These are almost-sure conditions, and no expectation requirements are imposed. For the case in which $X$ arises as the solution of a nonlinear stochastic integral equation driven by fractional Brownian motion, we provide a simple criterion on the drift coefficient under which the law of $X$ is automatically equivalent to the one of fractional Brownian motion. |
| title | A criterion for absolute continuity relative to the law of fractional Brownian motion |
| topic | Probability |
| url | https://arxiv.org/abs/2306.11824 |