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).

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).

On the first hitting time and the last exit time for a Brownian motion to/from a moving boundary

1988 ◽  
Vol 20 (2) ◽  
pp. 411-426 ◽  
Author(s):  
Paavo Salminen
Keyword(s):  
Brownian Motion ◽  
Smooth Function ◽  
Moving Boundary ◽  
Exit Time ◽  
Expected Value ◽  
Explicit Solutions ◽  
First Hitting Time ◽  
Standard Brownian Motion ◽  
Square Root ◽  
Last Exit Time

Let t → h(t) be a smooth function on ℝ+, and B = {Bs; s ≥ 0} a standard Brownian motion. In this paper we derive expressions for the distributions of the variables Th: = inf {S; Bs = h(s)} and λth: = sup {s ≦ t; Bs = h(s)}, where t> 0 is given. Our formulas contain an expected value of a Brownian functional. It is seen that this can be computed, principally, using Feynman–Kac&s formula. Further, we discuss in our framework the familiar examples with linear and square root boundaries. Moreover our approach provides in some extent explicit solutions for the second-order boundaries.

Quasi-stationary distributions for a Brownian motion with drift and associated limit laws

1994 ◽  
Vol 31 (4) ◽  
pp. 911-920 ◽  
Author(s):  
Servet Martinez ◽  
Jaime San Martin
Keyword(s):  
Brownian Motion ◽  
Limit Distribution ◽  
Invariant Measures ◽  
Three Dimensional ◽  
Transition Density ◽  
Bessel Process ◽  
Limit Laws ◽  
Brownian Motion With Drift ◽  
Infinite Mass ◽  
Negative Drift

We prove that the quasi-invariant measures associated to a Brownian motion with negative drift X form a one-parameter family. The minimal one is a probability measure inducing the transition density of a three-dimensional Bessel process, and it is shown that it is the density of the limit distribution limt→∞Px(X A | τ > t). It is also shown that the minimal quasi-invariant measure of infinite mass induces the density of the limit distribution ) which is the law of a Bessel process with drift.

On the first hitting time and the last exit time for a Brownian motion to/from a moving boundary

1988 ◽  
Vol 20 (02) ◽  
pp. 411-426 ◽  
Author(s):  
Paavo Salminen
Keyword(s):  
Brownian Motion ◽  
Smooth Function ◽  
Moving Boundary ◽  
Exit Time ◽  
Expected Value ◽  
Explicit Solutions ◽  
First Hitting Time ◽  
Standard Brownian Motion ◽  
Square Root ◽  
Last Exit Time

Lett → h(t) be a smooth function on ℝ+, andB= {Bs;s≥ 0} a standard Brownian motion. In this paper we derive expressions for the distributions of the variablesTh: = inf {S;Bs=h(s)} and λth: = sup {s≦t; Bs= h(s)}, wheret>0 is given. Our formulas contain an expected value of a Brownian functional. It is seen that this can be computed, principally, using Feynman–Kac&s formula. Further, we discuss in our framework the familiar examples with linear and square root boundaries. Moreover our approach provides in some extent explicit solutions for the second-order boundaries.

Quasi-stationary distributions for a Brownian motion with drift and associated limit laws

1994 ◽  
Vol 31 (04) ◽  
pp. 911-920 ◽  
Author(s):  
Servet Martinez ◽  
Jaime San Martin
Keyword(s):  
Brownian Motion ◽  
Limit Distribution ◽  
Invariant Measures ◽  
Three Dimensional ◽  
Transition Density ◽  
Bessel Process ◽  
Limit Laws ◽  
Brownian Motion With Drift ◽  
Infinite Mass ◽  
Negative Drift

We prove that the quasi-invariant measures associated to a Brownian motion with negative drift X form a one-parameter family. The minimal one is a probability measure inducing the transition density of a three-dimensional Bessel process, and it is shown that it is the density of the limit distribution lim t→∞ P x (X A | τ > t). It is also shown that the minimal quasi-invariant measure of infinite mass induces the density of the limit distribution ) which is the law of a Bessel process with drift.

Domain of attraction of quasi-stationary distributions for the brownian motion with drift

1998 ◽  
Vol 30 (2) ◽  
pp. 385-408 ◽  
Author(s):  
Servet Martinez ◽  
Pierre Picco ◽  
Jaime San Martin
Keyword(s):  
Brownian Motion ◽  
Stationary Distribution ◽  
Domain Of Attraction ◽  
Sufficient Condition ◽  
Stationary Distributions ◽  
Brownian Motion With Drift ◽  
Counter Example ◽  
Negative Drift ◽  
Initial Measure ◽  
Quasi Stationary Distribution

We consider Brownian motion with a negative drift conditioned to stay positive. We give a sufficient condition for an initial measure to be in the domain of attraction of a quasi-stationary distribution. We construct a counter-example that strongly suggests that this condition is optimal.

Domain of attraction of quasi-stationary distributions for the brownian motion with drift

1998 ◽  
Vol 30 (02) ◽  
pp. 385-408 ◽  
Author(s):  
Servet Martinez ◽  
Pierre Picco ◽  
Jaime San Martin
Keyword(s):  
Brownian Motion ◽  
Stationary Distribution ◽  
Domain Of Attraction ◽  
Sufficient Condition ◽  
Stationary Distributions ◽  
Brownian Motion With Drift ◽  
Counter Example ◽  
Negative Drift ◽  
Initial Measure ◽  
Quasi Stationary Distribution

We consider Brownian motion with a negative drift conditioned to stay positive. We give a sufficient condition for an initial measure to be in the domain of attraction of a quasi-stationary distribution. We construct a counter-example that strongly suggests that this condition is optimal.

On wald-type optimal stopping for Brownian motion

1997 ◽  
Vol 34 (1) ◽  
pp. 66-73 ◽  
Author(s):  
S. E. Graversen ◽  
G. Peškir
Keyword(s):  
Brownian Motion ◽  
Optimal Stopping ◽  
Stopping Time ◽  
Maximal Inequality ◽  
Stopping Times ◽  
Standard Brownian Motion ◽  
Square Root ◽  
Real Analysis ◽  
Optimal Stopping Problems ◽  
Wald's Identity

The solution is presented to all optimal stopping problems of the form supτE(G(|Β τ |) – cτ), where is standard Brownian motion and the supremum is taken over all stopping times τ for B with finite expectation, while the map G : ℝ+ → ℝ satisfies for some being given and fixed. The optimal stopping time is shown to be the hitting time by the reflecting Brownian motion of the set of all (approximate) maximum points of the map . The method of proof relies upon Wald's identity for Brownian motion and simple real analysis arguments. A simple proof of the Dubins–Jacka–Schwarz–Shepp–Shiryaev (square root of two) maximal inequality for randomly stopped Brownian motion is given as an application.

Approximations of boundary crossing probabilities for a Brownian motion

1999 ◽  
Vol 36 (4) ◽  
pp. 1019-1030 ◽  
Author(s):  
Alex Novikov ◽  
Volf Frishling ◽  
Nino Kordzakhia
Keyword(s):  
Brownian Motion ◽  
Power Series ◽  
Numerical Integration ◽  
Piecewise Linear ◽  
Boundary Crossing ◽  
Square Root ◽  
Girsanov Transformation ◽  
Piecewise Linear Approximations ◽  
Infinite Power ◽  
Crossing Probabilities

Using the Girsanov transformation we derive estimates for the accuracy of piecewise approximations for one-sided and two-sided boundary crossing probabilities. We demonstrate that piecewise linear approximations can be calculated using repeated numerical integration. As an illustrative example we consider the case of one-sided and two-sided square-root boundaries for which we also present analytical representations in a form of infinite power series.

Optimal Control of a Model for a System Subject to Continuous Wear

1988 ◽  
Vol 2 (3) ◽  
pp. 321-328 ◽  
Author(s):  
Laurence A. Baxter ◽  
Eui Yong Lee
Keyword(s):  
Optimal Control ◽  
Brownian Motion ◽  
Poisson Process ◽  
Average Cost ◽  
Infinite Horizon ◽  
Arrival Rate ◽  
The State ◽  
Negative Drift ◽  
Random Amount

The state of a system is modelled by Brownian motion with negative drift and an absorbing barrier at the origin. A repairman arrives according to a Poisson process of rate λ. If the state of the system at arrival of the repairman does not exceed a certain threshold, he/she increases it by a random amount, otherwise no action is taken. Costs are assigned to each visit of the repairman, to each repair, and to the system being in state 0. It is shown that there exists a unique arrival rate λ which minimizes the average cost per unit time over an infinite horizon.

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