Boundary non-crossing probabilities for fractional Brownian motion with trend

Abstract

In this paper, we investigate the boundary non-crossing probabilities of a fractional Brownian motion considering some general deterministic trend function. We derive bounds for non-crossing probabilities and discuss the case of a large trend function. As a by-product, we solve a minimization problem related to the norm of the trend function.

Keywords::

• boundary crossings
• Cameron–Martin–Girsanov theorem
• reproducing kernel Hilbert space
• large deviation principle
• Molchan martingale
• fractional Brownian motion

Acknowledgements

We are in debt to two referees and an Associate Editor for several suggestions which improved our manuscript. E. Hashorva and Y. Mishura acknowledge support from the SNSF Grants 200021-140633/1, 200020-159246/1 and the project RARE -318984 (an FP7 Marie Curie IRSES Fellowship).

Notes

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