A criterion for absolute continuity relative to the law of fractional Brownian motion
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Bibliographic record
Abstract
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.
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| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.000 |
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| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
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