| Issue |
ESAIM: PS
Volume 29, 2025
|
|
|---|---|---|
| Page(s) | 357 - 380 | |
| DOI | https://doi.org/10.1051/ps/2025010 | |
| Published online | 05 August 2025 | |
Drawdowns of Diffusions
1
AAbo Akademi University Faculty of Science and Engineering
20500
AAbo,
Finland
2
Université de Lorraine CNRS, INRIA, IECL
F-54000
Nancy,
France
* Corresponding author: pierre.vallois@univ-lorraine.fr
Received:
29
November
2024
Accepted:
12
May
2025
In this paper, using the excursion theory, we provide new proofs of, firstly, Lehoczky’s formula (in an extended form allowing a lower bound for the underlying diffusion) for the joint distribution of the first drawdown time and the maximum before this time, and, secondly, of Malyutin’s formula for the joint distribution of the first hitting time and the maximum drawdown before this time. It is remarkable – but there is a clean explanation – that the excursion theoretical approach which we developed first for Lehoczky’s formula also provides a proof for Malyutin’s formula. Moreover, we discuss some generalizations and analyze the pure jump process describing the maximum before the first drawdown time when the size of the drawdown is varying.
Mathematics Subject Classification: 60J60 / 60J65 / 60G40
Key words: Brownian motion / Poisson point process / additive functional / local time / pure jump process
© The authors. Published by EDP Sciences, SMAI 2025
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