Summary
We study how Brownian motion behaves under time change by a fluctuating additive functionalA t , in particular letting τ be the first passage time ofA t to zero we computeP −x[B τ ∈dy] explicitly in certain cases. The calculation is not an easy one, our method uses the Désiré André relation for the overshoot of a Lévy process and depends on some elliptic function identities. This paper only considers the one boundary case whereA t is increasing (resp. decreasing) on the positive (resp. negative) half line.
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McGill, P. Wiener-Hopf factorisation of Brownian motion. Probab. Th. Rel. Fields 83, 355–389 (1989). https://doi.org/10.1007/BF00964370
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DOI: https://doi.org/10.1007/BF00964370