Electronic Communications in Probability

The maximal drawdown of the Brownian meander

Yueyun Hu, Zhan Shi, and Marc Yor

Full-text: Open access

Abstract

Motivated by evaluating the limiting distribution of randomly biased random walks on trees, we compute the exact value of a negative moment of the maximal drawdown of the standard Brownian meander.

Article information

Source
Electron. Commun. Probab., Volume 20 (2015), paper no. 39, 6 pp.

Dates
Accepted: 20 May 2015
First available in Project Euclid: 7 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465320966

Digital Object Identifier
doi:10.1214/ECP.v20-3945

Mathematical Reviews number (MathSciNet)
MR3352334

Zentralblatt MATH identifier
1325.60134

Subjects
Primary: 60J65: Brownian motion [See also 58J65]

Keywords
Brownian meander Bessel process maximal drawdown

Rights
This work is licensed under a Creative Commons Attribution 3.0 License.

Citation

Hu, Yueyun; Shi, Zhan; Yor, Marc. The maximal drawdown of the Brownian meander. Electron. Commun. Probab. 20 (2015), paper no. 39, 6 pp. doi:10.1214/ECP.v20-3945. https://projecteuclid.org/euclid.ecp/1465320966


Export citation

References

  • Carraro, Laurent; El Karoui, Nicole; ObłÃ³j, Jan. On Azéma-Yor processes, their optimal properties and the Bachelier-drawdown equation. Ann. Probab. 40 (2012), no. 1, 372–400.
  • Cheridito, Patrick; Nikeghbali, Ashkan; Platen, Eckhard. Processes of class sigma, last passage times, and drawdowns. SIAM J. Financial Math. 3 (2012), no. 1, 280–303.
  • Cherny, Vladimir; ObłÃ³j, Jan. Portfolio optimisation under non-linear drawdown constraints in a semimartingale financial model. Finance Stoch. 17 (2013), no. 4, 771–800.
  • Hu, Y. and Shi, Z. (2014+). Localizing biased random walks on trees. (Preprint available on http://www.math.univ-paris13.fr/~yueyun/yzlocaltree.pdf)
  • Imhof, J.-P. Density factorizations for Brownian motion, meander and the three-dimensional Bessel process, and applications. J. Appl. Probab. 21 (1984), no. 3, 500–510.
  • Lehoczky, John P. Formulas for stopped diffusion processes with stopping times based on the maximum. Ann. Probability 5 (1977), no. 4, 601–607.
  • Mijatović, Aleksandar; Pistorius, Martijn R. On the drawdown of completely asymmetric Lévy processes. Stochastic Process. Appl. 122 (2012), no. 11, 3812–3836.
  • Nikeghbali, Ashkan. A class of remarkable submartingales. Stochastic Process. Appl. 116 (2006), no. 6, 917–938.
  • Revuz, Daniel; Yor, Marc. Continuous martingales and Brownian motion. Third edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 293. Springer-Verlag, Berlin, 1999. xiv+602 pp. ISBN: 3-540-64325-7
  • Rieder, Ulrich; Wittlinger, Marc. On optimal terminal wealth problems with random trading times and drawdown constraints. Adv. in Appl. Probab. 46 (2014), no. 1, 121–138.
  • Yen, Ju-Yi; Yor, Marc. Local times and excursion theory for Brownian motion. A tale of Wiener and It� measures. Lecture Notes in Mathematics, 2088. Springer, Cham, 2013. x+135 pp. ISBN: 978-3-319-01269-8; 978-3-319-01270-4
  • Zhang, Hongzhong; Hadjiliadis, Olympia. Drawdowns and rallies in a finite time-horizon. Drawdowns and rallies. Methodol. Comput. Appl. Probab. 12 (2010), no. 2, 293–308.