Brownian Motion Path and Maximum Drawdown

2007 ◽  
Keyword(s):  
Brownian Motion ◽  
Motion Path ◽  
Maximum Drawdown

scholarly journals On the maximum drawdown of a Brownian motion

2004 ◽  
Vol 41 (1) ◽  
pp. 147-161 ◽  
Author(s):  
Malik Magdon-Ismail ◽  
Amir F. Atiya ◽  
Amrit Pratap ◽  
Yaser S. Abu-Mostafa
Keyword(s):  
Brownian Motion ◽  
Random Process ◽  
Infinite Series ◽  
Series Representation ◽  
Expected Value ◽  
Square Root ◽  
Brownian Motion With Drift ◽  
Negative Drift ◽  
Analytic Expression ◽  
Maximum Drawdown

The maximum drawdown at time T of a random process on [0,T] can be defined informally as the largest drop from a peak to a trough. In this paper, we investigate the behaviour of this statistic for a Brownian motion with drift. In particular, we give an infinite series representation of its distribution and consider its expected value. When the drift is zero, we give an analytic expression for the expected value, and for nonzero drift, we give an infinite series representation. For all cases, we compute the limiting (T → ∞) behaviour, which can be logarithmic (for positive drift), square root (for zero drift) or linear (for negative drift).

The measure theory of random fractals

1986 ◽  
Vol 100 (3) ◽  
pp. 383-406 ◽  
Author(s):  
S. James Taylor
Keyword(s):  
Brownian Motion ◽  
Potential Theory ◽  
Measure Theory ◽  
Motion Path ◽  
Natural Question ◽  
Random Fractals ◽  
Limited Space ◽  
Definition Of ◽  
Good Problem

In 1951 A. S. Besicovitch, who was my research supervisor, suggested that I look at the problem of determining the dimension of the range of a Brownian motion path. This problem had been communicated to him by C. Loewner, but it was a natural question which had already attracted the attention of Paul Lévy. It was a good problem to give to an ignorant Ph.D. student because it forced him to learn the potential theory of Frostman [33] and Riesz[75] as well as the Wiener [98] definition of mathematical Brownian motion. In fact the solution of that first problem in [81] used only ideas which were already twenty-five years old, though at the time they seemed both new and original to me. My purpose in this paper is to try to trace the development of these techniques as they have been exploited by many authors and used in diverse situations since 1953. As we do this in the limited space available it will be impossible to even outline all aspects of the development, so I make no apology for giving a biased account concentrating on those areas of most interest to me. At the same time I will make conjectures and suggest some problems which are natural and accessible in the hope of stimulating further research.

Super-Brownian motion: Path properties and hitting probabilities

1989 ◽  
Vol 83 (1-2) ◽  
pp. 135-205 ◽  
Author(s):  
D. A. Dawson ◽  
I. Iscoe ◽  
E. A. Perkins
Keyword(s):  
Brownian Motion ◽  
Motion Path ◽  
Hitting Probabilities ◽  
Super Brownian Motion ◽  
Path Properties

scholarly journals On the maximum drawdown of a Brownian motion

2004 ◽  
Vol 41 (01) ◽  
pp. 147-161 ◽  
Author(s):  
Malik Magdon-Ismail ◽  
Amir F. Atiya ◽  
Amrit Pratap ◽  
Yaser S. Abu-Mostafa
Keyword(s):  
Brownian Motion ◽  
Random Process ◽  
Infinite Series ◽  
Series Representation ◽  
Expected Value ◽  
Square Root ◽  
Brownian Motion With Drift ◽  
Negative Drift ◽  
Analytic Expression ◽  
Maximum Drawdown

The maximum drawdown at time T of a random process on [0,T] can be defined informally as the largest drop from a peak to a trough. In this paper, we investigate the behaviour of this statistic for a Brownian motion with drift. In particular, we give an infinite series representation of its distribution and consider its expected value. When the drift is zero, we give an analytic expression for the expected value, and for nonzero drift, we give an infinite series representation. For all cases, we compute the limiting (T → ∞) behaviour, which can be logarithmic (for positive drift), square root (for zero drift) or linear (for negative drift).

Integrated Fractional white Noise as an Alternative to Multifractional Brownian Motion

2007 ◽  
Vol 44 (02) ◽  
pp. 393-408 ◽  
Author(s):  
Allan Sly
Keyword(s):  
Stochastic Process ◽  
Brownian Motion ◽  
White Noise ◽  
Gaussian Process ◽  
Discrete Data ◽  
Hölder Exponent ◽  
Scaling Properties ◽  
Multifractional Brownian Motion ◽  
Local Properties

Multifractional Brownian motion is a Gaussian process which has changing scaling properties generated by varying the local Hölder exponent. We show that multifractional Brownian motion is very sensitive to changes in the selected Hölder exponent and has extreme changes in magnitude. We suggest an alternative stochastic process, called integrated fractional white noise, which retains the important local properties but avoids the undesirable oscillations in magnitude. We also show how the Hölder exponent can be estimated locally from discrete data in this model.

Brownian motion with quadratic killing and some implications

1986 ◽  
Vol 23 (04) ◽  
pp. 893-903 ◽  
Author(s):  
Michael L. Wenocur
Keyword(s):  
Brownian Motion ◽  
Weibull Distribution ◽  
Special Function ◽  
Function Theory ◽  
Killing Rate

Brownian motion subject to a quadratic killing rate and its connection with the Weibull distribution is analyzed. The distribution obtained for the process killing time significantly generalizes the Weibull. The derivation involves the use of the Karhunen–Loève expansion for Brownian motion, special function theory, and the calculus of residues.

Model demonstration of brownian motion

1971 ◽  
Vol 105 (12) ◽  
pp. 736-736
Author(s):  
V.I. Arabadzhi
Keyword(s):  
Brownian Motion
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